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Search, Evaluate, and Read Evidence","\u002Fen\u002Facademic-writing\u002F02-reading-and-literature-matrix","en\u002Facademic-writing\u002F02-reading-and-literature-matrix",{"title":27,"path":28,"stem":29},"3. From Sources to Synthesis and Argument","\u002Fen\u002Facademic-writing\u002F03-arguments-outlines-and-paragraphs","en\u002Facademic-writing\u002F03-arguments-outlines-and-paragraphs",{"title":31,"path":32,"stem":33},"4. Literature Review and a Defensible Gap","\u002Fen\u002Facademic-writing\u002F04-writing-the-literature-review","en\u002Facademic-writing\u002F04-writing-the-literature-review",{"title":35,"path":36,"stem":37},"5. Research Design, Evidence, and Core Sections","\u002Fen\u002Facademic-writing\u002F05-core-sections-and-evidence","en\u002Facademic-writing\u002F05-core-sections-and-evidence",{"title":39,"path":40,"stem":41},"6. Citation, Paraphrasing, Integrity, and AI","\u002Fen\u002Facademic-writing\u002F06-citation-paraphrasing-and-integrity","en\u002Facademic-writing\u002F06-citation-paraphrasing-and-integrity",{"title":43,"path":44,"stem":45},"7. Revision, Review, and Submission","\u002Fen\u002Facademic-writing\u002F07-revision-style-and-submission","en\u002Facademic-writing\u002F07-revision-style-and-submission",{"title":47,"path":48,"stem":49},"8. Research Workbook","\u002Fen\u002Facademic-writing\u002F08-literature-review-checklist-and-template","en\u002Facademic-writing\u002F08-literature-review-checklist-and-template",{"title":51,"path":52,"stem":53},"9. Research Workflow and Tools","\u002Fen\u002Facademic-writing\u002F09-tools-for-academic-writing-and-literature-management","en\u002Facademic-writing\u002F09-tools-for-academic-writing-and-literature-management",{"title":55,"path":56,"stem":57},"10. Worked Project — From Question to Defensible Conclusion","\u002Fen\u002Facademic-writing\u002F10-worked-example-from-block-structure-to-question-chain","en\u002Facademic-writing\u002F10-worked-example-from-block-structure-to-question-chain",{"title":59,"path":60,"stem":61,"children":62,"page":249},"Accounting","\u002Fen\u002Faccounting","en\u002Faccounting",[63,69,87,93,193],{"title":64,"path":65,"stem":66,"children":67},"Accounting — From Evidence to Decisions","\u002Fen\u002Faccounting\u002F00-index","en\u002Faccounting\u002F00-index",[68],{"title":64,"path":65,"stem":66},{"title":70,"path":71,"stem":72,"children":73},"Accounting Appendix","\u002Fen\u002Faccounting\u002Fappendix","en\u002Faccounting\u002Fappendix\u002Findex",[74,75,79,83],{"title":70,"path":71,"stem":72},{"title":76,"path":77,"stem":78},"Worked Examples and Error Diagnosis","\u002Fen\u002Faccounting\u002Fappendix\u002F24-examples-pitfalls","en\u002Faccounting\u002Fappendix\u002F24-examples-pitfalls",{"title":80,"path":81,"stem":82},"Accounting Glossary","\u002Fen\u002Faccounting\u002Fappendix\u002F26-glossary","en\u002Faccounting\u002Fappendix\u002F26-glossary",{"title":84,"path":85,"stem":86},"Reading and Evidence Map","\u002Fen\u002Faccounting\u002Fappendix\u002F27-reading-map","en\u002Faccounting\u002Fappendix\u002F27-reading-map",{"title":88,"path":89,"stem":90,"children":91},"Northstar Record-to-Decision Capstone","\u002Fen\u002Faccounting\u002Fcapstone","en\u002Faccounting\u002Fcapstone\u002Findex",[92],{"title":88,"path":89,"stem":90},{"title":94,"path":95,"stem":96,"children":97},"Financial Accounting","\u002Fen\u002Faccounting\u002Ffinancial-accounting","en\u002Faccounting\u002Ffinancial-accounting\u002Findex",[98,99,117,131,165,179],{"title":94,"path":95,"stem":96},{"title":100,"path":101,"stem":102,"children":103},"1. Foundations","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002Findex",[104,105,109,113],{"title":100,"path":101,"stem":102},{"title":106,"path":107,"stem":108},"Objectives and Qualitative Characteristics","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F01-objectives-qualitative-characteristics","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F01-objectives-qualitative-characteristics",{"title":110,"path":111,"stem":112},"Equation and Elements","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F02-equation-elements","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F02-equation-elements",{"title":114,"path":115,"stem":116},"Accrual Basis, Estimates and Periods","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F03-accounting-bases-principles","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F03-accounting-bases-principles",{"title":118,"path":119,"stem":120,"children":121},"2. Recording Transactions","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions","en\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002Findex",[122,123,127],{"title":118,"path":119,"stem":120},{"title":124,"path":125,"stem":126},"Double-Entry and Debit\u002FCredit","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F04-double-entry-dr-cr","en\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F04-double-entry-dr-cr",{"title":128,"path":129,"stem":130},"Accounting Cycle","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F05-accounting-cycle","en\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F05-accounting-cycle",{"title":132,"path":133,"stem":134,"children":135},"3. Measurement and Adjustments","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002Findex",[136,137,141,145,149,153,157,161],{"title":132,"path":133,"stem":134},{"title":138,"path":139,"stem":140},"Revenue Recognition","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F07-revenue-recognition","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F07-revenue-recognition",{"title":142,"path":143,"stem":144},"Inventory","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F08-inventory","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F08-inventory",{"title":146,"path":147,"stem":148},"Receivables and Expected Credit Losses","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F09-receivables","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F09-receivables",{"title":150,"path":151,"stem":152},"Property, Plant and Equipment","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F10-ppe","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F10-ppe",{"title":154,"path":155,"stem":156},"Intangible Assets","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F11-intangibles","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F11-intangibles",{"title":158,"path":159,"stem":160},"Leases","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F12-leases","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F12-leases",{"title":162,"path":163,"stem":164},"Current and Deferred Income Tax","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F13-income-tax","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F13-income-tax",{"title":166,"path":167,"stem":168,"children":169},"4. Reporting and Cash","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash","en\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002Findex",[170,171,175],{"title":166,"path":167,"stem":168},{"title":172,"path":173,"stem":174},"Linked Financial Statements","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F06-financial-statements","en\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F06-financial-statements",{"title":176,"path":177,"stem":178},"Cash, Reconciliation and Internal Control","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F14-cash-control","en\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F14-cash-control",{"title":180,"path":181,"stem":182,"children":183},"5. Analysis and Comparison","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison","en\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002Findex",[184,185,189],{"title":180,"path":181,"stem":182},{"title":186,"path":187,"stem":188},"Ratio Analysis","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F15-ratio-analysis","en\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F15-ratio-analysis",{"title":190,"path":191,"stem":192},"IFRS versus US GAAP","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F25-ifrs-gaap","en\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F25-ifrs-gaap",{"title":194,"path":195,"stem":196,"children":197},"Management Accounting","\u002Fen\u002Faccounting\u002Fmanagement-accounting","en\u002Faccounting\u002Fmanagement-accounting\u002Findex",[198,199,217,235],{"title":194,"path":195,"stem":196},{"title":200,"path":201,"stem":202,"children":203},"1. Cost Foundations","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002Findex",[204,205,209,213],{"title":200,"path":201,"stem":202},{"title":206,"path":207,"stem":208},"Cost Concepts and Behaviour","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F16-cost-concepts","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F16-cost-concepts",{"title":210,"path":211,"stem":212},"Costing Systems","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F17-costing-systems","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F17-costing-systems",{"title":214,"path":215,"stem":216},"Variable and Absorption Costing","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F19-variable-vs-absorption","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F19-variable-vs-absorption",{"title":218,"path":219,"stem":220,"children":221},"2. Planning and Control","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002Findex",[222,223,227,231],{"title":218,"path":219,"stem":220},{"title":224,"path":225,"stem":226},"Cost–Volume–Profit Analysis","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F18-cvp-analysis","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F18-cvp-analysis",{"title":228,"path":229,"stem":230},"Budgeting and Variance Analysis","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F20-budgeting-variances","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F20-budgeting-variances",{"title":232,"path":233,"stem":234},"Performance Measurement","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F21-performance-measurement","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F21-performance-measurement",{"title":236,"path":237,"stem":238,"children":239},"3. Decisions and Investment","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment","en\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002Findex",[240,241,245],{"title":236,"path":237,"stem":238},{"title":242,"path":243,"stem":244},"Short-Term Decisions","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F22-short-term-decisions","en\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F22-short-term-decisions",{"title":246,"path":247,"stem":248},"Capital Budgeting","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F23-capital-budgeting","en\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F23-capital-budgeting",false,{"title":251,"path":252,"stem":253,"children":254,"page":249},"Business Analytics","\u002Fen\u002Fbusiness-analytics","en\u002Fbusiness-analytics",[255,261,279,305,335,365,383,400],{"title":256,"path":257,"stem":258,"children":259},"Business Analytics — From Data to Defensible Action","\u002Fen\u002Fbusiness-analytics\u002F00-index","en\u002Fbusiness-analytics\u002F00-index",[260],{"title":256,"path":257,"stem":258},{"title":262,"path":263,"stem":264,"children":265},"0. Decision and Data Foundations","\u002Fen\u002Fbusiness-analytics\u002F00-intro","en\u002Fbusiness-analytics\u002F00-intro\u002Findex",[266,267,271,275],{"title":262,"path":263,"stem":264},{"title":268,"path":269,"stem":270},"Decision Framing","\u002Fen\u002Fbusiness-analytics\u002F00-intro\u002F01-decision-framing","en\u002Fbusiness-analytics\u002F00-intro\u002F01-decision-framing",{"title":272,"path":273,"stem":274},"Data Contracts","\u002Fen\u002Fbusiness-analytics\u002F00-intro\u002F02-data-contracts","en\u002Fbusiness-analytics\u002F00-intro\u002F02-data-contracts",{"title":276,"path":277,"stem":278},"Reproducible Analytics Workflow","\u002Fen\u002Fbusiness-analytics\u002F00-intro\u002F03-reproducible-workflow","en\u002Fbusiness-analytics\u002F00-intro\u002F03-reproducible-workflow",{"title":280,"path":281,"stem":282,"children":283},"1. Descriptive Analytics","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive","en\u002Fbusiness-analytics\u002F01-descriptive\u002Findex",[284,285,289,293,297,301],{"title":280,"path":281,"stem":282},{"title":286,"path":287,"stem":288},"Distributions and Exploratory Data Analysis","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F04-distributions-eda","en\u002Fbusiness-analytics\u002F01-descriptive\u002F04-distributions-eda",{"title":290,"path":291,"stem":292},"KPIs and Denominators","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F05-kpis-denominators","en\u002Fbusiness-analytics\u002F01-descriptive\u002F05-kpis-denominators",{"title":294,"path":295,"stem":296},"Segments, Cohorts and Funnels","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F06-segmentation-cohorts-funnels","en\u002Fbusiness-analytics\u002F01-descriptive\u002F06-segmentation-cohorts-funnels",{"title":298,"path":299,"stem":300},"Visual Evidence and Data Stories","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F07-visualisation-story","en\u002Fbusiness-analytics\u002F01-descriptive\u002F07-visualisation-story",{"title":302,"path":303,"stem":304},"Experiments and Causal Boundaries","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F08-experiments-causal-boundary","en\u002Fbusiness-analytics\u002F01-descriptive\u002F08-experiments-causal-boundary",{"title":306,"path":307,"stem":308,"children":309},"2. Predictive Analytics","\u002Fen\u002Fbusiness-analytics\u002F02-predictive","en\u002Fbusiness-analytics\u002F02-predictive\u002Findex",[310,311,315,319,323,327,331],{"title":306,"path":307,"stem":308},{"title":312,"path":313,"stem":314},"Validation, Baselines and Leakage","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F09-validation-leakage","en\u002Fbusiness-analytics\u002F02-predictive\u002F09-validation-leakage",{"title":316,"path":317,"stem":318},"Regression and Forecasting","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F10-regression-forecasting","en\u002Fbusiness-analytics\u002F02-predictive\u002F10-regression-forecasting",{"title":320,"path":321,"stem":322},"Classification and Calibration","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F11-classification-calibration","en\u002Fbusiness-analytics\u002F02-predictive\u002F11-classification-calibration",{"title":324,"path":325,"stem":326},"Trees and Ensembles","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F12-trees-ensembles","en\u002Fbusiness-analytics\u002F02-predictive\u002F12-trees-ensembles",{"title":328,"path":329,"stem":330},"Decision Metrics and Thresholds","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F13-decision-metrics-thresholds","en\u002Fbusiness-analytics\u002F02-predictive\u002F13-decision-metrics-thresholds",{"title":332,"path":333,"stem":334},"Explainability, Monitoring and Drift","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F14-explainability-drift","en\u002Fbusiness-analytics\u002F02-predictive\u002F14-explainability-drift",{"title":336,"path":337,"stem":338,"children":339},"3. Prescriptive Analytics","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive","en\u002Fbusiness-analytics\u002F03-prescriptive\u002Findex",[340,341,345,349,353,357,361],{"title":336,"path":337,"stem":338},{"title":342,"path":343,"stem":344},"Decision-Making under Uncertainty","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F15-decision-uncertainty","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F15-decision-uncertainty",{"title":346,"path":347,"stem":348},"Linear Optimisation","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F16-linear-optimization","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F16-linear-optimization",{"title":350,"path":351,"stem":352},"Inventory and Allocation","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F17-inventory-allocation","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F17-inventory-allocation",{"title":354,"path":355,"stem":356},"Queueing and Simulation","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F18-queueing-simulation","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F18-queueing-simulation",{"title":358,"path":359,"stem":360},"Pricing and Revenue Management","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F19-pricing-experimentation","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F19-pricing-experimentation",{"title":362,"path":363,"stem":364},"Causal Targeting and Policy Learning","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F20-causal-targeting","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F20-causal-targeting",{"title":366,"path":367,"stem":368,"children":369},"4. Deployment and Governance","\u002Fen\u002Fbusiness-analytics\u002F04-deployment","en\u002Fbusiness-analytics\u002F04-deployment\u002Findex",[370,371,375,379],{"title":366,"path":367,"stem":368},{"title":372,"path":373,"stem":374},"Data Products and Monitoring","\u002Fen\u002Fbusiness-analytics\u002F04-deployment\u002F21-data-products-monitoring","en\u002Fbusiness-analytics\u002F04-deployment\u002F21-data-products-monitoring",{"title":376,"path":377,"stem":378},"Governance, Fairness and Privacy","\u002Fen\u002Fbusiness-analytics\u002F04-deployment\u002F22-governance-fairness-privacy","en\u002Fbusiness-analytics\u002F04-deployment\u002F22-governance-fairness-privacy",{"title":380,"path":381,"stem":382},"Adoption and Business Value","\u002Fen\u002Fbusiness-analytics\u002F04-deployment\u002F23-adoption-value","en\u002Fbusiness-analytics\u002F04-deployment\u002F23-adoption-value",{"title":384,"path":385,"stem":386,"children":387},"Appendix and Revision Tools","\u002Fen\u002Fbusiness-analytics\u002Fappendix","en\u002Fbusiness-analytics\u002Fappendix\u002Findex",[388,389,393,397],{"title":384,"path":385,"stem":386},{"title":390,"path":391,"stem":392},"Formula and Decision Map","\u002Fen\u002Fbusiness-analytics\u002Fappendix\u002F24-formula-map","en\u002Fbusiness-analytics\u002Fappendix\u002F24-formula-map",{"title":394,"path":395,"stem":396},"Worked Examples and Pitfalls","\u002Fen\u002Fbusiness-analytics\u002Fappendix\u002F25-worked-examples-pitfalls","en\u002Fbusiness-analytics\u002Fappendix\u002F25-worked-examples-pitfalls",{"title":84,"path":398,"stem":399},"\u002Fen\u002Fbusiness-analytics\u002Fappendix\u002F26-reading-map","en\u002Fbusiness-analytics\u002Fappendix\u002F26-reading-map",{"title":401,"path":402,"stem":403,"children":404},"Capstone — The Evening Delivery Promise","\u002Fen\u002Fbusiness-analytics\u002Fcapstone","en\u002Fbusiness-analytics\u002Fcapstone\u002Findex",[405],{"title":401,"path":402,"stem":403},{"title":407,"path":408,"stem":409,"children":410,"page":249},"Financial Economic Time Series","\u002Fen\u002Ffinancial-economic-time-series","en\u002Ffinancial-economic-time-series",[411,417,423,429,435,441,447,453,459,465,471],{"title":412,"path":413,"stem":414,"children":415},"Financial and Economic Time Series — Course Guide","\u002Fen\u002Ffinancial-economic-time-series\u002F00-intro","en\u002Ffinancial-economic-time-series\u002F00-intro\u002Findex",[416],{"title":412,"path":413,"stem":414},{"title":418,"path":419,"stem":420,"children":421},"Three Versions of Time Series","\u002Fen\u002Ffinancial-economic-time-series\u002F01-bridge","en\u002Ffinancial-economic-time-series\u002F01-bridge\u002Findex",[422],{"title":418,"path":419,"stem":420},{"title":424,"path":425,"stem":426,"children":427},"Data, Clocks, and Transformations","\u002Fen\u002Ffinancial-economic-time-series\u002F02-data-transformations","en\u002Ffinancial-economic-time-series\u002F02-data-transformations\u002Findex",[428],{"title":424,"path":425,"stem":426},{"title":430,"path":431,"stem":432,"children":433},"Predictive Regressions and Persistent Predictors","\u002Fen\u002Ffinancial-economic-time-series\u002F03-predictive-regressions","en\u002Ffinancial-economic-time-series\u002F03-predictive-regressions\u002Findex",[434],{"title":430,"path":431,"stem":432},{"title":436,"path":437,"stem":438,"children":439},"Volatility, Tails, and Financial Risk","\u002Fen\u002Ffinancial-economic-time-series\u002F04-volatility-risk","en\u002Ffinancial-economic-time-series\u002F04-volatility-risk\u002Findex",[440],{"title":436,"path":437,"stem":438},{"title":442,"path":443,"stem":444,"children":445},"Unit Roots, Cointegration, and Error Correction","\u002Fen\u002Ffinancial-economic-time-series\u002F05-unit-roots-cointegration","en\u002Ffinancial-economic-time-series\u002F05-unit-roots-cointegration\u002Findex",[446],{"title":442,"path":443,"stem":444},{"title":448,"path":449,"stem":450,"children":451},"VAR, Structural Identification, and Local Projections","\u002Fen\u002Ffinancial-economic-time-series\u002F06-var-identification","en\u002Ffinancial-economic-time-series\u002F06-var-identification\u002Findex",[452],{"title":448,"path":449,"stem":450},{"title":454,"path":455,"stem":456,"children":457},"State Space, Mixed Frequency, and Nowcasting","\u002Fen\u002Ffinancial-economic-time-series\u002F07-state-space-nowcasting","en\u002Ffinancial-economic-time-series\u002F07-state-space-nowcasting\u002Findex",[458],{"title":454,"path":455,"stem":456},{"title":460,"path":461,"stem":462,"children":463},"Forecast Evaluation for Decisions","\u002Fen\u002Ffinancial-economic-time-series\u002F08-forecast-evaluation","en\u002Ffinancial-economic-time-series\u002F08-forecast-evaluation\u002Findex",[464],{"title":460,"path":461,"stem":462},{"title":466,"path":467,"stem":468,"children":469},"Integrated R Laboratory","\u002Fen\u002Ffinancial-economic-time-series\u002F09-r-laboratory","en\u002Ffinancial-economic-time-series\u002F09-r-laboratory\u002Findex",[470],{"title":466,"path":467,"stem":468},{"title":472,"path":473,"stem":474,"children":475},"Capstones, Data, and Reading Ladder","\u002Fen\u002Ffinancial-economic-time-series\u002F10-capstone-readings","en\u002Ffinancial-economic-time-series\u002F10-capstone-readings\u002Findex",[476],{"title":472,"path":473,"stem":474},{"title":478,"path":479,"stem":480,"children":481,"page":249},"Intro To Economics","\u002Fen\u002Fintro-to-economics","en\u002Fintro-to-economics",[482,486,490,494,498,502,506,510,514,518,522,526,530],{"title":483,"path":484,"stem":485},"Introduction to Economics — Decisions, Markets, and the Macroeconomy","\u002Fen\u002Fintro-to-economics\u002F00-intro","en\u002Fintro-to-economics\u002F00-intro",{"title":487,"path":488,"stem":489},"Chapter 1 — Choice, Opportunity Cost, and Trade","\u002Fen\u002Fintro-to-economics\u002F01-foundations","en\u002Fintro-to-economics\u002F01-foundations",{"title":491,"path":492,"stem":493},"Chapter 2 — Demand, Supply, Equilibrium, and Welfare","\u002Fen\u002Fintro-to-economics\u002F02-demand-and-supply","en\u002Fintro-to-economics\u002F02-demand-and-supply",{"title":495,"path":496,"stem":497},"Chapter 3 — Elasticity, Revenue, and Tax Incidence","\u002Fen\u002Fintro-to-economics\u002F03-elasticity","en\u002Fintro-to-economics\u002F03-elasticity",{"title":499,"path":500,"stem":501},"Chapter 4 — Firms, Market Power, and Market Failure","\u002Fen\u002Fintro-to-economics\u002F04-market-structures","en\u002Fintro-to-economics\u002F04-market-structures",{"title":503,"path":504,"stem":505},"Chapter 5 — GDP, Income, Wealth, and Welfare","\u002Fen\u002Fintro-to-economics\u002F05-gdp-and-wealth","en\u002Fintro-to-economics\u002F05-gdp-and-wealth",{"title":507,"path":508,"stem":509},"Chapter 6 — Inflation, Purchasing Power, and Labour Markets","\u002Fen\u002Fintro-to-economics\u002F06-inflation-and-unemployment","en\u002Fintro-to-economics\u002F06-inflation-and-unemployment",{"title":511,"path":512,"stem":513},"Chapter 7 — Productivity, Technology, and Economic Growth","\u002Fen\u002Fintro-to-economics\u002F07-economic-growth","en\u002Fintro-to-economics\u002F07-economic-growth",{"title":515,"path":516,"stem":517},"Chapter 8 — Money, Credit, and Banking","\u002Fen\u002Fintro-to-economics\u002F08-money-and-banking","en\u002Fintro-to-economics\u002F08-money-and-banking",{"title":519,"path":520,"stem":521},"Chapter 9 — Business Cycles, AD–AS, and Monetary Policy","\u002Fen\u002Fintro-to-economics\u002F09-monetary-policy-and-ad-as","en\u002Fintro-to-economics\u002F09-monetary-policy-and-ad-as",{"title":523,"path":524,"stem":525},"Chapter 10 — Fiscal Policy, Distribution, and Public Debt","\u002Fen\u002Fintro-to-economics\u002F10-fiscal-policy","en\u002Fintro-to-economics\u002F10-fiscal-policy",{"title":527,"path":528,"stem":529},"Chapter 11 — Trade, Capital Flows, and Exchange Rates","\u002Fen\u002Fintro-to-economics\u002F11-open-economy-and-exchange-rates","en\u002Fintro-to-economics\u002F11-open-economy-and-exchange-rates",{"title":531,"path":532,"stem":533},"Chapter 12 — Integrated Economic Analysis Studio","\u002Fen\u002Fintro-to-economics\u002F12-review-and-case-studies","en\u002Fintro-to-economics\u002F12-review-and-case-studies",{"title":535,"path":536,"stem":537,"children":538,"page":249},"Microeconometrics","\u002Fen\u002Fmicroeconometrics","en\u002Fmicroeconometrics",[539,557,563,581,603,629,651,673,691,709],{"title":540,"path":541,"stem":542,"children":543},"0. Causal Questions and Designs","\u002Fen\u002Fmicroeconometrics\u002F00-foundations","en\u002Fmicroeconometrics\u002F00-foundations\u002Findex",[544,545,549,553],{"title":540,"path":541,"stem":542},{"title":546,"path":547,"stem":548},"Causal Questions and Estimands","\u002Fen\u002Fmicroeconometrics\u002F00-foundations\u002F01-causal-question-estimands","en\u002Fmicroeconometrics\u002F00-foundations\u002F01-causal-question-estimands",{"title":550,"path":551,"stem":552},"Potential Outcomes and Experiments","\u002Fen\u002Fmicroeconometrics\u002F00-foundations\u002F02-potential-outcomes-experiments","en\u002Fmicroeconometrics\u002F00-foundations\u002F02-potential-outcomes-experiments",{"title":554,"path":555,"stem":556},"Causal Diagrams and Controls","\u002Fen\u002Fmicroeconometrics\u002F00-foundations\u002F03-dags-controls","en\u002Fmicroeconometrics\u002F00-foundations\u002F03-dags-controls",{"title":558,"path":559,"stem":560,"children":561},"Microeconometrics — Designing Credible Counterfactuals","\u002Fen\u002Fmicroeconometrics\u002F00-index","en\u002Fmicroeconometrics\u002F00-index",[562],{"title":558,"path":559,"stem":560},{"title":564,"path":565,"stem":566,"children":567},"1. Regression and Inference","\u002Fen\u002Fmicroeconometrics\u002F01-regression","en\u002Fmicroeconometrics\u002F01-regression\u002Findex",[568,569,573,577],{"title":564,"path":565,"stem":566},{"title":570,"path":571,"stem":572},"OLS as a Projection","\u002Fen\u002Fmicroeconometrics\u002F01-regression\u002F04-ols-projection","en\u002Fmicroeconometrics\u002F01-regression\u002F04-ols-projection",{"title":574,"path":575,"stem":576},"FWL, Selection and Controls","\u002Fen\u002Fmicroeconometrics\u002F01-regression\u002F05-fwl-selection-controls","en\u002Fmicroeconometrics\u002F01-regression\u002F05-fwl-selection-controls",{"title":578,"path":579,"stem":580},"Inference and Clustering","\u002Fen\u002Fmicroeconometrics\u002F01-regression\u002F06-inference-clustering","en\u002Fmicroeconometrics\u002F01-regression\u002F06-inference-clustering",{"title":582,"path":583,"stem":584,"children":585},"2. Instruments and Panel Data","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel","en\u002Fmicroeconometrics\u002F02-iv-panel\u002Findex",[586,587,591,595,599],{"title":582,"path":583,"stem":584},{"title":588,"path":589,"stem":590},"Instrumental Variables","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F07-iv-identification","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F07-iv-identification",{"title":592,"path":593,"stem":594},"Weak Instruments and LATE","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F08-weak-iv-late","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F08-weak-iv-late",{"title":596,"path":597,"stem":598},"Panel Fixed Effects","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F09-panel-fixed-effects","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F09-panel-fixed-effects",{"title":600,"path":601,"stem":602},"Dynamic Panels and Limits","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F10-dynamic-panel","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F10-dynamic-panel",{"title":604,"path":605,"stem":606,"children":607},"3. Policy Evaluation Designs","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs","en\u002Fmicroeconometrics\u002F03-policy-designs\u002Findex",[608,609,613,617,621,625],{"title":604,"path":605,"stem":606},{"title":610,"path":611,"stem":612},"Difference-in-Differences","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F11-did-core","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F11-did-core",{"title":614,"path":615,"stem":616},"Staggered DID and Event Studies","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F12-staggered-event-studies","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F12-staggered-event-studies",{"title":618,"path":619,"stem":620},"Regression Discontinuity","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F13-rdd","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F13-rdd",{"title":622,"path":623,"stem":624},"Matching, Weighting and Overlap","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F14-matching-weighting","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F14-matching-weighting",{"title":626,"path":627,"stem":628},"Synthetic Control","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F15-synthetic-control","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F15-synthetic-control",{"title":630,"path":631,"stem":632,"children":633},"Module 4 — Choice and Limited Outcomes","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002Findex",[634,635,639,643,647],{"title":630,"path":631,"stem":632},{"title":636,"path":637,"stem":638},"Binary Choice","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F16-binary-choice","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F16-binary-choice",{"title":640,"path":641,"stem":642},"Multinomial and Ordered Choice","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F17-multinomial-choice","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F17-multinomial-choice",{"title":644,"path":645,"stem":646},"Count Outcomes","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F18-count-outcomes","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F18-count-outcomes",{"title":648,"path":649,"stem":650},"Censoring, Truncation and Selection","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F19-censoring-selection","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F19-censoring-selection",{"title":652,"path":653,"stem":654,"children":655},"Module 5 — Modern Causal Analysis","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal","en\u002Fmicroeconometrics\u002F05-modern-causal\u002Findex",[656,657,661,665,669],{"title":652,"path":653,"stem":654},{"title":658,"path":659,"stem":660},"Double Machine Learning","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F20-dml","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F20-dml",{"title":662,"path":663,"stem":664},"Heterogeneity and Policy Learning","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F21-heterogeneity-policy","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F21-heterogeneity-policy",{"title":666,"path":667,"stem":668},"Sensitivity and Partial Identification","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F22-sensitivity-partial-id","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F22-sensitivity-partial-id",{"title":670,"path":671,"stem":672},"External Validity and Transport","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F23-external-validity","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F23-external-validity",{"title":674,"path":675,"stem":676,"children":677},"Module 6 — From Data to Auditable Evidence","\u002Fen\u002Fmicroeconometrics\u002F06-workflow","en\u002Fmicroeconometrics\u002F06-workflow\u002Findex",[678,679,683,687],{"title":674,"path":675,"stem":676},{"title":680,"path":681,"stem":682},"Data Provenance","\u002Fen\u002Fmicroeconometrics\u002F06-workflow\u002F24-data-provenance","en\u002Fmicroeconometrics\u002F06-workflow\u002F24-data-provenance",{"title":684,"path":685,"stem":686},"Reproducibility and Reporting","\u002Fen\u002Fmicroeconometrics\u002F06-workflow\u002F25-reproducibility-reporting","en\u002Fmicroeconometrics\u002F06-workflow\u002F25-reproducibility-reporting",{"title":688,"path":689,"stem":690},"Paper and Evidence Audit","\u002Fen\u002Fmicroeconometrics\u002F06-workflow\u002F26-paper-audit","en\u002Fmicroeconometrics\u002F06-workflow\u002F26-paper-audit",{"title":692,"path":693,"stem":694,"children":695},"Appendix — Revision and Evidence Map","\u002Fen\u002Fmicroeconometrics\u002Fappendix","en\u002Fmicroeconometrics\u002Fappendix\u002Findex",[696,697,701,705],{"title":692,"path":693,"stem":694},{"title":698,"path":699,"stem":700},"Formula and Design Map","\u002Fen\u002Fmicroeconometrics\u002Fappendix\u002F27-formula-design-map","en\u002Fmicroeconometrics\u002Fappendix\u002F27-formula-design-map",{"title":702,"path":703,"stem":704},"Ten Worked Microeconometric Pitfalls","\u002Fen\u002Fmicroeconometrics\u002Fappendix\u002F28-worked-pitfalls","en\u002Fmicroeconometrics\u002Fappendix\u002F28-worked-pitfalls",{"title":706,"path":707,"stem":708},"Reading and Software Map","\u002Fen\u002Fmicroeconometrics\u002Fappendix\u002F29-reading-software-map","en\u002Fmicroeconometrics\u002Fappendix\u002F29-reading-software-map",{"title":710,"path":711,"stem":712,"children":713},"Capstone — Should Northbridge Expand Pathways?","\u002Fen\u002Fmicroeconometrics\u002Fcapstone","en\u002Fmicroeconometrics\u002Fcapstone\u002Findex",[714],{"title":710,"path":711,"stem":712},{"title":716,"path":717,"stem":718,"children":719,"page":249},"Microeconomics","\u002Fen\u002Fmicroeconomics","en\u002Fmicroeconomics",[720,726,732,738,744,750,756,762,768,774,780,786,792],{"title":721,"path":722,"stem":723,"children":724},"Advanced Microeconomics — Course Guide","\u002Fen\u002Fmicroeconomics\u002F00-intro","en\u002Fmicroeconomics\u002F00-intro\u002Findex",[725],{"title":721,"path":722,"stem":723},{"title":727,"path":728,"stem":729,"children":730},"Module 1 — Choice, Duality, and Revealed Preference","\u002Fen\u002Fmicroeconomics\u002F01-fundations","en\u002Fmicroeconomics\u002F01-fundations\u002Findex",[731],{"title":727,"path":728,"stem":729},{"title":733,"path":734,"stem":735,"children":736},"Module 2 — Comparative Statics and Welfare Measurement","\u002Fen\u002Fmicroeconomics\u002F02-comparative-statics","en\u002Fmicroeconomics\u002F02-comparative-statics\u002Findex",[737],{"title":733,"path":734,"stem":735},{"title":739,"path":740,"stem":741,"children":742},"Module 3 — Choice under Risk and Insurance","\u002Fen\u002Fmicroeconomics\u002F03-uncertainty","en\u002Fmicroeconomics\u002F03-uncertainty\u002Findex",[743],{"title":739,"path":740,"stem":741},{"title":745,"path":746,"stem":747,"children":748},"Module 4 — General Equilibrium and Welfare","\u002Fen\u002Fmicroeconomics\u002F04-general-equilibrium","en\u002Fmicroeconomics\u002F04-general-equilibrium\u002Findex",[749],{"title":745,"path":746,"stem":747},{"title":751,"path":752,"stem":753,"children":754},"Module 5 — Static, Dynamic, and Repeated Games","\u002Fen\u002Fmicroeconomics\u002F05-game-theory","en\u002Fmicroeconomics\u002F05-game-theory\u002Findex",[755],{"title":751,"path":752,"stem":753},{"title":757,"path":758,"stem":759,"children":760},"Module 6 — Oligopoly, Entry, and Algorithmic Pricing","\u002Fen\u002Fmicroeconomics\u002F06-oligopoly","en\u002Fmicroeconomics\u002F06-oligopoly\u002Findex",[761],{"title":757,"path":758,"stem":759},{"title":763,"path":764,"stem":765,"children":766},"Module 7 — Information Economics and Contracts","\u002Fen\u002Fmicroeconomics\u002F07-information-economics","en\u002Fmicroeconomics\u002F07-information-economics\u002Findex",[767],{"title":763,"path":764,"stem":765},{"title":769,"path":770,"stem":771,"children":772},"Module 8 — Mechanism Design and Auctions","\u002Fen\u002Fmicroeconomics\u002F08-mechanism-design","en\u002Fmicroeconomics\u002F08-mechanism-design\u002Findex",[773],{"title":769,"path":770,"stem":771},{"title":775,"path":776,"stem":777,"children":778},"Module 9 — Behavioural and Experimental Microeconomics","\u002Fen\u002Fmicroeconomics\u002F09-behavioural-economics","en\u002Fmicroeconomics\u002F09-behavioural-economics\u002Findex",[779],{"title":775,"path":776,"stem":777},{"title":781,"path":782,"stem":783,"children":784},"Module 10 — Externalities, Public Goods, and Collective Action","\u002Fen\u002Fmicroeconomics\u002F10-externalities-public-goods","en\u002Fmicroeconomics\u002F10-externalities-public-goods\u002Findex",[785],{"title":781,"path":782,"stem":783},{"title":787,"path":788,"stem":789,"children":790},"Module 11 — Matching and Market Design","\u002Fen\u002Fmicroeconomics\u002F11-market-design","en\u002Fmicroeconomics\u002F11-market-design\u002Findex",[791],{"title":787,"path":788,"stem":789},{"title":793,"path":794,"stem":795,"children":796},"Module 12 — Integrated Microeconomic Design Studio","\u002Fen\u002Fmicroeconomics\u002F12-review","en\u002Fmicroeconomics\u002F12-review\u002Findex",[797],{"title":793,"path":794,"stem":795},{"title":799,"path":800,"stem":801,"children":802},"Playground","\u002Fen\u002Fplayground","en\u002Fplayground\u002Findex",[803,804,808,812,816,820],{"title":799,"path":800,"stem":801},{"title":805,"path":806,"stem":807},"Typst Plugin Playground","\u002Fen\u002Fplayground\u002F01-typstex","en\u002Fplayground\u002F01-typstex",{"title":809,"path":810,"stem":811},"Citation Plugin Test","\u002Fen\u002Fplayground\u002F02-citation","en\u002Fplayground\u002F02-citation",{"title":813,"path":814,"stem":815},"Pyodide Playground","\u002Fen\u002Fplayground\u002F03-pyodide","en\u002Fplayground\u002F03-pyodide",{"title":817,"path":818,"stem":819},"WebR Playground","\u002Fen\u002Fplayground\u002F04-webr","en\u002Fplayground\u002F04-webr",{"title":821,"path":822,"stem":823},"Chart.js 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文献综述的构建与写作","\u002Fzh\u002Facademic-writing\u002F04-writing-the-literature-review","zh\u002Facademic-writing\u002F04-writing-the-literature-review",{"title":1056,"path":1057,"stem":1058},"8. 文献综述清单与模板","\u002Fzh\u002Facademic-writing\u002F08-literature-review-checklist-and-template","zh\u002Facademic-writing\u002F08-literature-review-checklist-and-template",{"title":1060,"path":1061,"stem":1062},"10. 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",[1797,1798,1801,1841],"span",{"className":1799},[1800],"katex",[1797,1802,1805],{"className":1803},[1804],"katex-mathml",[1806,1807,1809],"math",{"xmlns":1808},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[1810,1811,1812,1836],"semantics",{},[1813,1814,1815,1819,1824,1827,1830,1833],"mrow",{},[1816,1817,1818],"mi",{},"P",[1820,1821,1823],"mo",{"stretchy":1822},"false","(",[1816,1825,1826],{},"X",[1820,1828,1829],{},">",[1816,1831,1832],{},"x",[1820,1834,1835],{"stretchy":1822},")",[1837,1838,1840],"annotation",{"encoding":1839},"application\u002Fx-tex","P(X>x)",[1797,1842,1846,1881],{"className":1843,"ariaHidden":1845},[1844],"katex-html","true",[1797,1847,1850,1855,1861,1865,1869,1874,1878],{"className":1848},[1849],"base",[1797,1851],{"className":1852,"style":1854},[1853],"strut","height:1em;vertical-align:-0.25em;",[1797,1856,1818],{"className":1857,"style":1860},[1858,1859],"mord","mathnormal","margin-right:0.1389em;",[1797,1862,1823],{"className":1863},[1864],"mopen",[1797,1866,1826],{"className":1867,"style":1868},[1858,1859],"margin-right:0.0785em;",[1797,1870],{"className":1871,"style":1873},[1872],"mspace","margin-right:0.2778em;",[1797,1875,1829],{"className":1876},[1877],"mrel",[1797,1879],{"className":1880,"style":1873},[1872],[1797,1882,1884,1887,1890],{"className":1883},[1849],[1797,1885],{"className":1886,"style":1854},[1853],[1797,1888,1832],{"className":1889},[1858,1859],[1797,1891,1835],{"className":1892},[1893],"mclose"," decay at large ",[1797,1896,1898,1911],{"className":1897},[1800],[1797,1899,1901],{"className":1900},[1804],[1806,1902,1903],{"xmlns":1808},[1810,1904,1905,1909],{},[1813,1906,1907],{},[1816,1908,1832],{},[1837,1910,1832],{"encoding":1839},[1797,1912,1914],{"className":1913,"ariaHidden":1845},[1844],[1797,1915,1917,1921],{"className":1916},[1849],[1797,1918],{"className":1919,"style":1920},[1853],"height:0.4306em;",[1797,1922,1832],{"className":1923},[1858,1859],"? It matters when a few losses dominate reserves, reinsurance, or capital.",[1926,1927,1929],"h2",{"id":1928},"_1-compare-tail-classes","1. Compare tail classes",[1931,1932,1933,1952],"table",{},[1934,1935,1936],"thead",{},[1937,1938,1939,1943,1946,1949],"tr",{},[1940,1941,1942],"th",{},"Model",[1940,1944,1945],{},"Survival behaviour",[1940,1947,1948],{},"Moment implication",[1940,1950,1951],{},"Typical 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positive moments finite",[1958,2093,2094],{},"flexible light\u002Fstretch-exponential tail",[1937,2096,2097,2100,2103,2106],{},[1958,2098,2099],{},"Lognormal",[1958,2101,2102],{},"slower than any exponential, faster than any power",[1958,2104,2105],{},"all moments finite",[1958,2107,2108],{},"multiplicative severity benchmark",[1937,2110,2111,2114,2239,2325],{},[1958,2112,2113],{},"Pareto\u002FLomax",[1958,2115,2116,2117],{},"power law 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",[1797,2243,2245,2259],{"className":2244},[1800],[1797,2246,2248],{"className":2247},[1804],[1806,2249,2250],{"xmlns":1808},[1810,2251,2252,2257],{},[1813,2253,2254],{},[1816,2255,2256],{},"r",[1837,2258,2256],{"encoding":1839},[1797,2260,2262],{"className":2261,"ariaHidden":1845},[1844],[1797,2263,2265,2268],{"className":2264},[1849],[1797,2266],{"className":2267,"style":1920},[1853],[1797,2269,2256],{"className":2270,"style":2201},[1858,1859]," exists only if ",[1797,2273,2275,2294],{"className":2274},[1800],[1797,2276,2278],{"className":2277},[1804],[1806,2279,2280],{"xmlns":1808},[1810,2281,2282,2291],{},[1813,2283,2284,2286,2289],{},[1816,2285,2256],{},[1820,2287,2288],{},"\u003C",[1816,2290,2155],{},[1837,2292,2293],{"encoding":1839},"r\u003C\\alpha",[1797,2295,2297,2316],{"className":2296,"ariaHidden":1845},[1844],[1797,2298,2300,2304,2307,2310,2313],{"className":2299},[1849],[1797,2301],{"className":2302,"style":2303},[1853],"height:0.5782em;vertical-align:-0.0391em;",[1797,2305,2256],{"className":2306,"style":2201},[1858,1859],[1797,2308],{"className":2309,"style":1873},[1872],[1797,2311,2288],{"className":2312},[1877],[1797,2314],{"className":2315,"style":1873},[1872],[1797,2317,2319,2322],{"className":2318},[1849],[1797,2320],{"className":2321,"style":1920},[1853],[1797,2323,2155],{"className":2324,"style":2238},[1858,1859],[1958,2326,2327],{},"heavy-tail benchmark",[1937,2329,2330,2333,2369,2424],{},[1958,2331,2332],{},"GPD exceedance",[1958,2334,2335,2336,2368],{},"shape ",[1797,2337,2339,2354],{"className":2338},[1800],[1797,2340,2342],{"className":2341},[1804],[1806,2343,2344],{"xmlns":1808},[1810,2345,2346,2351],{},[1813,2347,2348],{},[1816,2349,2350],{},"ξ",[1837,2352,2353],{"encoding":1839},"\\xi",[1797,2355,2357],{"className":2356,"ariaHidden":1845},[1844],[1797,2358,2360,2364],{"className":2359},[1849],[1797,2361],{"className":2362,"style":2363},[1853],"height:0.8889em;vertical-align:-0.1944em;",[1797,2365,2350],{"className":2366,"style":2367},[1858,1859],"margin-right:0.046em;"," determines tail class",[1958,2370,2371,2372],{},"mean exists if ",[1797,2373,2375,2393],{"className":2374},[1800],[1797,2376,2378],{"className":2377},[1804],[1806,2379,2380],{"xmlns":1808},[1810,2381,2382,2390],{},[1813,2383,2384,2386,2388],{},[1816,2385,2350],{},[1820,2387,2288],{},[2132,2389,2134],{},[1837,2391,2392],{"encoding":1839},"\\xi\u003C1",[1797,2394,2396,2414],{"className":2395,"ariaHidden":1845},[1844],[1797,2397,2399,2402,2405,2408,2411],{"className":2398},[1849],[1797,2400],{"className":2401,"style":2363},[1853],[1797,2403,2350],{"className":2404,"style":2367},[1858,1859],[1797,2406],{"className":2407,"style":1873},[1872],[1797,2409,2288],{"className":2410},[1877],[1797,2412],{"className":2413,"style":1873},[1872],[1797,2415,2417,2421],{"className":2416},[1849],[1797,2418],{"className":2419,"style":2420},[1853],"height:0.6444em;",[1797,2422,2134],{"className":2423},[1858],[1958,2425,2426],{},"losses above a high threshold",[1793,2428,2429],{},"“Heavy-tailed” has a mathematical meaning; it is not merely “large variance.”",[1926,2431,2433],{"id":2432},"_2-weibull-shape-controls-ageing","2. Weibull: shape controls ageing",[1793,2435,2436,2437,2490,2491,2542],{},"With scale ",[1797,2438,2440,2459],{"className":2439},[1800],[1797,2441,2443],{"className":2442},[1804],[1806,2444,2445],{"xmlns":1808},[1810,2446,2447,2456],{},[1813,2448,2449,2451,2453],{},[1816,2450,1997],{},[1820,2452,1829],{},[2132,2454,2455],{},"0",[1837,2457,2458],{"encoding":1839},"\\lambda>0",[1797,2460,2462,2481],{"className":2461,"ariaHidden":1845},[1844],[1797,2463,2465,2469,2472,2475,2478],{"className":2464},[1849],[1797,2466],{"className":2467,"style":2468},[1853],"height:0.7335em;vertical-align:-0.0391em;",[1797,2470,1997],{"className":2471},[1858,1859],[1797,2473],{"className":2474,"style":1873},[1872],[1797,2476,1829],{"className":2477},[1877],[1797,2479],{"className":2480,"style":1873},[1872],[1797,2482,2484,2487],{"className":2483},[1849],[1797,2485],{"className":2486,"style":2420},[1853],[1797,2488,2455],{"className":2489},[1858]," and shape ",[1797,2492,2494,2512],{"className":2493},[1800],[1797,2495,2497],{"className":2496},[1804],[1806,2498,2499],{"xmlns":1808},[1810,2500,2501,2509],{},[1813,2502,2503,2505,2507],{},[1816,2504,2005],{},[1820,2506,1829],{},[2132,2508,2455],{},[1837,2510,2511],{"encoding":1839},"k>0",[1797,2513,2515,2533],{"className":2514,"ariaHidden":1845},[1844],[1797,2516,2518,2521,2524,2527,2530],{"className":2517},[1849],[1797,2519],{"className":2520,"style":2468},[1853],[1797,2522,2005],{"className":2523,"style":2085},[1858,1859],[1797,2525],{"className":2526,"style":1873},[1872],[1797,2528,1829],{"className":2529},[1877],[1797,2531],{"className":2532,"style":1873},[1872],[1797,2534,2536,2539],{"className":2535},[1849],[1797,2537],{"className":2538,"style":2420},[1853],[1797,2540,2455],{"className":2541},[1858],",",[1797,2544,2547],{"className":2545},[2546],"katex-display",[1797,2548,2550,2651],{"className":2549},[1800],[1797,2551,2553],{"className":2552},[1804],[1806,2554,2556],{"xmlns":1808,"display":2555},"block",[1810,2557,2558,2648],{},[1813,2559,2560,2569,2571,2573,2575,2578,2580,2582,2584,2586,2588,2590,2592,2594,2600,2602,2604,2607,2610,2612,2614,2616,2618,2625,2627,2629,2631,2633,2645],{},[2561,2562,2563,2566],"mover",{"accent":1845},[1816,2564,2565],{},"F",[1820,2567,2568],{},"ˉ",[1820,2570,1823],{"stretchy":1822},[1816,2572,1832],{},[1820,2574,1835],{"stretchy":1822},[1820,2576,2577],{},"=",[1816,2579,1977],{},[1820,2581,1980],{},[1820,2583,1983],{"stretchy":1822},[1820,2585,1986],{},[1820,2587,1823],{"stretchy":1822},[1816,2589,1832],{},[1816,2591,1994],{"mathvariant":1993},[1816,2593,1997],{},[1999,2595,2596,2598],{},[1820,2597,1835],{"stretchy":1822},[1816,2599,2005],{},[1820,2601,2008],{"stretchy":1822},[1820,2603,2542],{"separator":1845},[1872,2605],{"width":2606},"2em",[1816,2608,2609],{},"h",[1820,2611,1823],{"stretchy":1822},[1816,2613,1832],{},[1820,2615,1835],{"stretchy":1822},[1820,2617,2577],{},[2619,2620,2621,2623],"mfrac",{},[1816,2622,2005],{},[1816,2624,1997],{},[1820,2626,1823],{"stretchy":1822},[1816,2628,1832],{},[1816,2630,1994],{"mathvariant":1993},[1816,2632,1997],{},[1999,2634,2635,2637],{},[1820,2636,1835],{"stretchy":1822},[1813,2638,2639,2641,2643],{},[1816,2640,2005],{},[1820,2642,1986],{},[2132,2644,2134],{},[1816,2646,2647],{"mathvariant":1993},".",[1837,2649,2650],{"encoding":1839},"\\bar F(x)=\\exp[-(x\u002F\\lambda)^k],\n\\qquad\nh(x)=\\frac{k}{\\lambda}(x\u002F\\lambda)^{k-1}.",[1797,2652,2654,2717,2812],{"className":2653,"ariaHidden":1845},[1844],[1797,2655,2657,2661,2699,2702,2705,2708,2711,2714],{"className":2656},[1849],[1797,2658],{"className":2659,"style":2660},[1853],"height:1.0701em;vertical-align:-0.25em;",[1797,2662,2665],{"className":2663},[1858,2664],"accent",[1797,2666,2668],{"className":2667},[2057],[1797,2669,2671],{"className":2670},[2061],[1797,2672,2675,2685],{"className":2673,"style":2674},[2065],"height:0.8201em;",[1797,2676,2678,2682],{"style":2677},"top:-3em;",[1797,2679],{"className":2680,"style":2681},[2073],"height:3em;",[1797,2683,2565],{"className":2684,"style":1860},[1858,1859],[1797,2686,2688,2691],{"style":2687},"top:-3.2523em;",[1797,2689],{"className":2690,"style":2681},[2073],[1797,2692,2696],{"className":2693,"style":2695},[2694],"accent-body","left:-0.1667em;",[1797,2697,2568],{"className":2698},[1858],[1797,2700,1823],{"className":2701},[1864],[1797,2703,1832],{"className":2704},[1858,1859],[1797,2706,1835],{"className":2707},[1893],[1797,2709],{"className":2710,"style":1873},[1872],[1797,2712,2577],{"className":2713},[1877],[1797,2715],{"className":2716,"style":1873},[1872],[1797,2718,2720,2724,2727,2730,2733,2736,2739,2742,2745,2776,2779,2783,2787,2791,2794,2797,2800,2803,2806,2809],{"className":2719},[1849],[1797,2721],{"className":2722,"style":2723},[1853],"height:1.1491em;vertical-align:-0.25em;",[1797,2725,1977],{"className":2726},[2025],[1797,2728,1983],{"className":2729},[1864],[1797,2731,1986],{"className":2732},[1858],[1797,2734,1823],{"className":2735},[1864],[1797,2737,1832],{"className":2738},[1858,1859],[1797,2740,1994],{"className":2741},[1858],[1797,2743,1997],{"className":2744},[1858,1859],[1797,2746,2748,2751],{"className":2747},[1893],[1797,2749,1835],{"className":2750},[1893],[1797,2752,2754],{"className":2753},[2053],[1797,2755,2757],{"className":2756},[2057],[1797,2758,2760],{"className":2759},[2061],[1797,2761,2764],{"className":2762,"style":2763},[2065],"height:0.8991em;",[1797,2765,2767,2770],{"style":2766},"top:-3.113em;margin-right:0.05em;",[1797,2768],{"className":2769,"style":2074},[2073],[1797,2771,2773],{"className":2772},[2078,2079,2080,2081],[1797,2774,2005],{"className":2775,"style":2085},[1858,1859,2081],[1797,2777,2008],{"className":2778},[1893],[1797,2780,2542],{"className":2781},[2782],"mpunct",[1797,2784],{"className":2785,"style":2786},[1872],"margin-right:2em;",[1797,2788],{"className":2789,"style":2790},[1872],"margin-right:0.1667em;",[1797,2792,2609],{"className":2793},[1858,1859],[1797,2795,1823],{"className":2796},[1864],[1797,2798,1832],{"className":2799},[1858,1859],[1797,2801,1835],{"className":2802},[1893],[1797,2804],{"className":2805,"style":1873},[1872],[1797,2807,2577],{"className":2808},[1877],[1797,2810],{"className":2811,"style":1873},[1872],[1797,2813,2815,2819,2892,2895,2898,2901,2904,2942],{"className":2814},[1849],[1797,2816],{"className":2817,"style":2818},[1853],"height:2.0574em;vertical-align:-0.686em;",[1797,2820,2822,2826,2889],{"className":2821},[1858],[1797,2823],{"className":2824},[1864,2825],"nulldelimiter",[1797,2827,2829],{"className":2828},[2619],[1797,2830,2833,2880],{"className":2831},[2057,2832],"vlist-t2",[1797,2834,2836,2875],{"className":2835},[2061],[1797,2837,2840,2852,2863],{"className":2838,"style":2839},[2065],"height:1.3714em;",[1797,2841,2843,2846],{"style":2842},"top:-2.314em;",[1797,2844],{"className":2845,"style":2681},[2073],[1797,2847,2849],{"className":2848},[1858],[1797,2850,1997],{"className":2851},[1858,1859],[1797,2853,2855,2858],{"style":2854},"top:-3.23em;",[1797,2856],{"className":2857,"style":2681},[2073],[1797,2859],{"className":2860,"style":2862},[2861],"frac-line","border-bottom-width:0.04em;",[1797,2864,2866,2869],{"style":2865},"top:-3.677em;",[1797,2867],{"className":2868,"style":2681},[2073],[1797,2870,2872],{"className":2871},[1858],[1797,2873,2005],{"className":2874,"style":2085},[1858,1859],[1797,2876,2879],{"className":2877},[2878],"vlist-s","​",[1797,2881,2883],{"className":2882},[2061],[1797,2884,2887],{"className":2885,"style":2886},[2065],"height:0.686em;",[1797,2888],{},[1797,2890],{"className":2891},[1893,2825],[1797,2893,1823],{"className":2894},[1864],[1797,2896,1832],{"className":2897},[1858,1859],[1797,2899,1994],{"className":2900},[1858],[1797,2902,1997],{"className":2903},[1858,1859],[1797,2905,2907,2910],{"className":2906},[1893],[1797,2908,1835],{"className":2909},[1893],[1797,2911,2913],{"className":2912},[2053],[1797,2914,2916],{"className":2915},[2057],[1797,2917,2919],{"className":2918},[2061],[1797,2920,2922],{"className":2921,"style":2763},[2065],[1797,2923,2924,2927],{"style":2766},[1797,2925],{"className":2926,"style":2074},[2073],[1797,2928,2930],{"className":2929},[2078,2079,2080,2081],[1797,2931,2933,2936,2939],{"className":2932},[1858,2081],[1797,2934,2005],{"className":2935,"style":2085},[1858,1859,2081],[1797,2937,1986],{"className":2938},[2181,2081],[1797,2940,2134],{"className":2941},[1858,2081],[1797,2943,2647],{"className":2944},[1858],[2946,2947,2948,3004,3058],"ul",{},[2949,2950,2951,3003],"li",{},[1797,2952,2954,2972],{"className":2953},[1800],[1797,2955,2957],{"className":2956},[1804],[1806,2958,2959],{"xmlns":1808},[1810,2960,2961,2969],{},[1813,2962,2963,2965,2967],{},[1816,2964,2005],{},[1820,2966,2577],{},[2132,2968,2134],{},[1837,2970,2971],{"encoding":1839},"k=1",[1797,2973,2975,2994],{"className":2974,"ariaHidden":1845},[1844],[1797,2976,2978,2982,2985,2988,2991],{"className":2977},[1849],[1797,2979],{"className":2980,"style":2981},[1853],"height:0.6944em;",[1797,2983,2005],{"className":2984,"style":2085},[1858,1859],[1797,2986],{"className":2987,"style":1873},[1872],[1797,2989,2577],{"className":2990},[1877],[1797,2992],{"className":2993,"style":1873},[1872],[1797,2995,2997,3000],{"className":2996},[1849],[1797,2998],{"className":2999,"style":2420},[1853],[1797,3001,2134],{"className":3002},[1858],": Exponential, constant hazard;",[2949,3005,3006,3057],{},[1797,3007,3009,3027],{"className":3008},[1800],[1797,3010,3012],{"className":3011},[1804],[1806,3013,3014],{"xmlns":1808},[1810,3015,3016,3024],{},[1813,3017,3018,3020,3022],{},[1816,3019,2005],{},[1820,3021,2288],{},[2132,3023,2134],{},[1837,3025,3026],{"encoding":1839},"k\u003C1",[1797,3028,3030,3048],{"className":3029,"ariaHidden":1845},[1844],[1797,3031,3033,3036,3039,3042,3045],{"className":3032},[1849],[1797,3034],{"className":3035,"style":2468},[1853],[1797,3037,2005],{"className":3038,"style":2085},[1858,1859],[1797,3040],{"className":3041,"style":1873},[1872],[1797,3043,2288],{"className":3044},[1877],[1797,3046],{"className":3047,"style":1873},[1872],[1797,3049,3051,3054],{"className":3050},[1849],[1797,3052],{"className":3053,"style":2420},[1853],[1797,3055,2134],{"className":3056},[1858],": decreasing hazard;",[2949,3059,3060,3111],{},[1797,3061,3063,3081],{"className":3062},[1800],[1797,3064,3066],{"className":3065},[1804],[1806,3067,3068],{"xmlns":1808},[1810,3069,3070,3078],{},[1813,3071,3072,3074,3076],{},[1816,3073,2005],{},[1820,3075,1829],{},[2132,3077,2134],{},[1837,3079,3080],{"encoding":1839},"k>1",[1797,3082,3084,3102],{"className":3083,"ariaHidden":1845},[1844],[1797,3085,3087,3090,3093,3096,3099],{"className":3086},[1849],[1797,3088],{"className":3089,"style":2468},[1853],[1797,3091,2005],{"className":3092,"style":2085},[1858,1859],[1797,3094],{"className":3095,"style":1873},[1872],[1797,3097,1829],{"className":3098},[1877],[1797,3100],{"className":3101,"style":1873},[1872],[1797,3103,3105,3108],{"className":3104},[1849],[1797,3106],{"className":3107,"style":2420},[1853],[1797,3109,2134],{"className":3110},[1858],": increasing hazard.",[1793,3113,3114],{},"For claim size, hazard is a mathematical description of exceedance, not a literal failure rate unless the insurance mechanism supports that interpretation.",[1926,3116,3118],{"id":3117},"_3-pareto-iilomax-power-law-benchmark","3. Pareto II\u002FLomax: power-law benchmark",[1793,3120,3121],{},"This page 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F(x)=\\left(1+\\frac{x}{\\theta}\\right)^{-\\alpha},\n\\qquad x\\ge0, 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e":3795},[1853],"height:2.4271em;vertical-align:-0.936em;",[1797,3797,3799,3802,3954],{"className":3798},[1858],[1797,3800],{"className":3801},[1864,2825],[1797,3803,3805],{"className":3804},[2619],[1797,3806,3808,3945],{"className":3807},[2057,2832],[1797,3809,3811,3942],{"className":3810},[2061],[1797,3812,3815,3893,3901],{"className":3813,"style":3814},[2065],"height:1.4911em;",[1797,3816,3817,3820],{"style":2842},[1797,3818],{"className":3819,"style":2681},[2073],[1797,3821,3823,3826,3829,3832,3835,3838,3841,3872,3875,3878,3881,3884,3887,3890],{"className":3822},[1858],[1797,3824,1823],{"className":3825},[1864],[1797,3827,2155],{"className":3828,"style":2238},[1858,1859],[1797,3830],{"className":3831,"style":2177},[1872],[1797,3833,1986],{"className":3834},[2181],[1797,3836],{"className":3837,"style":2177},[1872],[1797,3839,2134],{"className":3840},[1858],[1797,3842,3844,3847],{"className":3843},[1893],[1797,3845,1835],{"className":3846},[1893],[1797,3848,3850],{"className":3849},[2053],[1797,3851,3853],{"className":3852},[2057],[1797,3854,3856],{"className":3855},[2061],[1797,3857,3860],{"className":3858,"style":3859},[2065],"height:0.7401em;",[1797,3861,3863,3866],{"style":3862},"top:-2.989em;margin-right:0.05em;",[1797,3864],{"className":3865,"style":2074},[2073],[1797,3867,3869],{"className":3868},[2078,2079,2080,2081],[1797,3870,3711],{"className":3871},[1858,2081],[1797,3873,1823],{"className":3874},[1864],[1797,3876,2155],{"className":3877,"style":2238},[1858,1859],[1797,3879],{"className":3880,"style":2177},[1872],[1797,3882,1986],{"className":3883},[2181],[1797,3885],{"className":3886,"style":2177},[1872],[1797,3888,3711],{"className":3889},[1858],[1797,3891,1835],{"className":3892},[1893],[1797,3894,3895,3898],{"style":2854},[1797,3896],{"className":3897,"style":2681},[2073],[1797,3899],{"className":3900,"style":2862},[2861],[1797,3902,3903,3906],{"style":2865},[1797,3904],{"className":3905,"style":2681},[2073],[1797,3907,3909,3912],{"className":3908},[1858],[1797,3910,2155],{"className":3911,"style":2238},[1858,1859],[1797,3913,3915,3918],{"className":3914},[1858],[1797,3916,2144],{"className":3917,"style":2201},[1858,1859],[1797,3919,3921],{"className":3920},[2053],[1797,3922,3924],{"className":3923},[2057],[1797,3925,3927],{"className":3926},[2061],[1797,3928,3931],{"className":3929,"style":3930},[2065],"height:0.8141em;",[1797,3932,3933,3936],{"style":2069},[1797,3934],{"className":3935,"style":2074},[2073],[1797,3937,3939],{"className":3938},[2078,2079,2080,2081],[1797,3940,3711],{"className":3941},[1858,2081],[1797,3943,2879],{"className":3944},[2878],[1797,3946,3948],{"className":3947},[2061],[1797,3949,3952],{"className":3950,"style":3951},[2065],"height:0.936em;",[1797,3953],{},[1797,3955],{"className":3956},[1893,2825],[1797,3958],{"className":3959,"style":3640},[1872],[1797,3961,1823],{"className":3962},[1864],[1797,3964,2155],{"className":3965,"style":2238},[1858,1859],[1797,3967],{"className":3968,"style":1873},[1872],[1797,3970,1829],{"className":3971},[1877],[1797,3973],{"className":3974,"style":1873},[1872],[1797,3976,3978,3981,3984,3987],{"className":3977},[1849],[1797,3979],{"className":3980,"style":1854},[1853],[1797,3982,3711],{"className":3983},[1858],[1797,3985,1835],{"className":3986},[1893],[1797,3988,2647],{"className":3989},[1858],[1793,3991,3992,3993,4045],{},"At ",[1797,3994,3996,4015],{"className":3995},[1800],[1797,3997,3999],{"className":3998},[1804],[1806,4000,4001],{"xmlns":1808},[1810,4002,4003,4012],{},[1813,4004,4005,4007,4009],{},[1816,4006,2155],{},[1820,4008,2577],{},[2132,4010,4011],{},"1.5",[1837,4013,4014],{"encoding":1839},"\\alpha=1.5",[1797,4016,4018,4036],{"className":4017,"ariaHidden":1845},[1844],[1797,4019,4021,4024,4027,4030,4033],{"className":4020},[1849],[1797,4022],{"className":4023,"style":1920},[1853],[1797,4025,2155],{"className":4026,"style":2238},[1858,1859],[1797,4028],{"className":4029,"style":1873},[1872],[1797,4031,2577],{"className":4032},[1877],[1797,4034],{"className":4035,"style":1873},[1872],[1797,4037,4039,4042],{"className":4038},[1849],[1797,4040],{"className":4041,"style":2420},[1853],[1797,4043,4011],{"className":4044},[1858],", the mean exists but variance is infinite. A finite sample still has a finite sample variance; the instability appears as the sample grows and new extremes arrive.",[4047,4048,4050,4051,4150,4151,4272,4273,4315],"tip",{"title":4049},"Parameterisation check","Some texts use a Pareto Type I distribution with support ",[1797,4052,4054,4078],{"className":4053},[1800],[1797,4055,4057],{"className":4056},[1804],[1806,4058,4059],{"xmlns":1808},[1810,4060,4061,4075],{},[1813,4062,4063,4065,4067],{},[1816,4064,1832],{},[1820,4066,3183],{},[4068,4069,4070,4072],"msub",{},[1816,4071,1832],{},[1816,4073,4074],{},"m",[1837,4076,4077],{"encoding":1839},"x\\ge x_m",[1797,4079,4081,4100],{"className":4080,"ariaHidden":1845},[1844],[1797,4082,4084,4088,4091,4094,4097],{"className":4083},[1849],[1797,4085],{"className":4086,"style":4087},[1853],"height:0.7719em;vertical-align:-0.136em;",[1797,4089,1832],{"className":4090},[1858,1859],[1797,4092],{"className":4093,"style":1873},[1872],[1797,4095,3183],{"className":4096},[1877],[1797,4098],{"className":4099,"style":1873},[1872],[1797,4101,4103,4107],{"className":4102},[1849],[1797,4104],{"className":4105,"style":4106},[1853],"height:0.5806em;vertical-align:-0.15em;",[1797,4108,4110,4113],{"className":4109},[1858],[1797,4111,1832],{"className":4112},[1858,1859],[1797,4114,4116],{"className":4115},[2053],[1797,4117,4119,4141],{"className":4118},[2057,2832],[1797,4120,4122,4138],{"className":4121},[2061],[1797,4123,4126],{"className":4124,"style":4125},[2065],"height:0.1514em;",[1797,4127,4129,4132],{"style":4128},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[1797,4130],{"className":4131,"style":2074},[2073],[1797,4133,4135],{"className":4134},[2078,2079,2080,2081],[1797,4136,4074],{"className":4137},[1858,1859,2081],[1797,4139,2879],{"className":4140},[2878],[1797,4142,4144],{"className":4143},[2061],[1797,4145,4148],{"className":4146,"style":4147},[2065],"height:0.15em;",[1797,4149],{}," and survival ",[1797,4152,4154,4184],{"className":4153},[1800],[1797,4155,4157],{"className":4156},[1804],[1806,4158,4159],{"xmlns":1808},[1810,4160,4161,4181],{},[1813,4162,4163,4165,4171,4173,4175],{},[1820,4164,1823],{"stretchy":1822},[4068,4166,4167,4169],{},[1816,4168,1832],{},[1816,4170,4074],{},[1816,4172,1994],{"mathvariant":1993},[1816,4174,1832],{},[1999,4176,4177,4179],{},[1820,4178,1835],{"stretchy":1822},[1816,4180,2155],{},[1837,4182,4183],{"encoding":1839},"(x_m\u002Fx)^\\alpha",[1797,4185,4187],{"className":4186,"ariaHidden":1845},[1844],[1797,4188,4190,4193,4196,4236,4239,4242],{"className":4189},[1849],[1797,4191],{"className":4192,"style":1854},[1853],[1797,4194,1823],{"className":4195},[1864],[1797,4197,4199,4202],{"className":4198},[1858],[1797,4200,1832],{"className":4201},[1858,1859],[1797,4203,4205],{"className":4204},[2053],[1797,4206,4208,4228],{"className":4207},[2057,2832],[1797,4209,4211,4225],{"className":4210},[2061],[1797,4212,4214],{"className":4213,"style":4125},[2065],[1797,4215,4216,4219],{"style":4128},[1797,4217],{"className":4218,"style":2074},[2073],[1797,4220,4222],{"className":4221},[2078,2079,2080,2081],[1797,4223,4074],{"className":4224},[1858,1859,2081],[1797,4226,2879],{"className":4227},[2878],[1797,4229,4231],{"className":4230},[2061],[1797,4232,4234],{"className":4233,"style":4147},[2065],[1797,4235],{},[1797,4237,1994],{"className":4238},[1858],[1797,4240,1832],{"className":4241},[1858,1859],[1797,4243,4245,4248],{"className":4244},[1893],[1797,4246,1835],{"className":4247},[1893],[1797,4249,4251],{"className":4250},[2053],[1797,4252,4254],{"className":4253},[2057],[1797,4255,4257],{"className":4256},[2061],[1797,4258,4261],{"className":4259,"style":4260},[2065],"height:0.6644em;",[1797,4262,4263,4266],{"style":2069},[1797,4264],{"className":4265,"style":2074},[2073],[1797,4267,4269],{"className":4268},[2078,2079,2080,2081],[1797,4270,2155],{"className":4271,"style":2238},[1858,1859,2081],". Always show the support and survival function rather than reporting only “Pareto(",[1797,4274,4276,4294],{"className":4275},[1800],[1797,4277,4279],{"className":4278},[1804],[1806,4280,4281],{"xmlns":1808},[1810,4282,4283,4291],{},[1813,4284,4285,4287,4289],{},[1816,4286,2155],{},[1820,4288,2542],{"separator":1845},[1816,4290,2144],{},[1837,4292,4293],{"encoding":1839},"\\alpha,\\theta",[1797,4295,4297],{"className":4296,"ariaHidden":1845},[1844],[1797,4298,4300,4303,4306,4309,4312],{"className":4299},[1849],[1797,4301],{"className":4302,"style":2363},[1853],[1797,4304,2155],{"className":4305,"style":2238},[1858,1859],[1797,4307,2542],{"className":4308},[2782],[1797,4310],{"className":4311,"style":2790},[1872],[1797,4313,2144],{"className":4314,"style":2201},[1858,1859],").”",[1926,4317,4319],{"id":4318},"_4-peaks-over-threshold-and-the-gpd","4. Peaks over threshold and the GPD",[1793,4321,4322,4323,4352,4353,4475],{},"Choose a high threshold ",[1797,4324,4326,4340],{"className":4325},[1800],[1797,4327,4329],{"className":4328},[1804],[1806,4330,4331],{"xmlns":1808},[1810,4332,4333,4338],{},[1813,4334,4335],{},[1816,4336,4337],{},"u",[1837,4339,4337],{"encoding":1839},[1797,4341,4343],{"className":4342,"ariaHidden":1845},[1844],[1797,4344,4346,4349],{"className":4345},[1849],[1797,4347],{"className":4348,"style":1920},[1853],[1797,4350,4337],{"className":4351},[1858,1859]," and define excess ",[1797,4354,4356,4388],{"className":4355},[1800],[1797,4357,4359],{"className":4358},[1804],[1806,4360,4361],{"xmlns":1808},[1810,4362,4363,4385],{},[1813,4364,4365,4368,4370,4372,4374,4376,4379,4381,4383],{},[1816,4366,4367],{},"Y",[1820,4369,2577],{},[1816,4371,1826],{},[1820,4373,1986],{},[1816,4375,4337],{},[1820,4377,4378],{},"∣",[1816,4380,1826],{},[1820,4382,1829],{},[1816,4384,4337],{},[1837,4386,4387],{"encoding":1839},"Y=X-u\\mid X>u",[1797,4389,4391,4410,4429,4447,4466],{"className":4390,"ariaHidden":1845},[1844],[1797,4392,4394,4398,4401,4404,4407],{"className":4393},[1849],[1797,4395],{"className":4396,"style":4397},[1853],"height:0.6833em;",[1797,4399,4367],{"className":4400,"style":2177},[1858,1859],[1797,4402],{"className":4403,"style":1873},[1872],[1797,4405,2577],{"className":4406},[1877],[1797,4408],{"className":4409,"style":1873},[1872],[1797,4411,4413,4417,4420,4423,4426],{"className":4412},[1849],[1797,4414],{"className":4415,"style":4416},[1853],"height:0.7667em;vertical-align:-0.0833em;",[1797,4418,1826],{"className":4419,"style":1868},[1858,1859],[1797,4421],{"className":4422,"style":2177},[1872],[1797,4424,1986],{"className":4425},[2181],[1797,4427],{"className":4428,"style":2177},[1872],[1797,4430,4432,4435,4438,4441,4444],{"className":4431},[1849],[1797,4433],{"className":4434,"style":1854},[1853],[1797,4436,4337],{"className":4437},[1858,1859],[1797,4439],{"className":4440,"style":1873},[1872],[1797,4442,4378],{"className":4443},[1877],[1797,4445],{"className":4446,"style":1873},[1872],[1797,4448,4450,4454,4457,4460,4463],{"className":4449},[1849],[1797,4451],{"className":4452,"style":4453},[1853],"height:0.7224em;vertical-align:-0.0391em;",[1797,4455,1826],{"className":4456,"style":1868},[1858,1859],[1797,4458],{"className":4459,"style":1873},[1872],[1797,4461,1829],{"className":4462},[1877],[1797,4464],{"className":4465,"style":1873},[1872],[1797,4467,4469,4472],{"className":4468},[1849],[1797,4470],{"className":4471,"style":1920},[1853],[1797,4473,4337],{"className":4474},[1858,1859],". Extreme-value theory motivates the Generalized Pareto distribution (GPD):",[1797,4477,4479],{"className":4478},[2546],[1797,4480,4482,4542],{"className":4481},[1800],[1797,4483,4485],{"className":4484},[1804],[1806,4486,4487],{"xmlns":1808,"display":2555},[1810,4488,4489,4539],{},[1813,4490,4491,4493,4495,4497,4499,4502,4504,4506,4537],{},[1816,4492,1818],{},[1820,4494,1823],{"stretchy":1822},[1816,4496,4367],{},[1820,4498,1829],{},[1816,4500,4501],{},"y",[1820,4503,1835],{"stretchy":1822},[1820,4505,2577],{},[1999,4507,4508,4527],{},[1813,4509,4510,4512,4514,4516,4518,4525],{},[1820,4511,1823],{"fence":1845},[2132,4513,2134],{},[1820,4515,2137],{},[1816,4517,2350],{},[2619,4519,4520,4522],{},[1816,4521,4501],{},[1816,4523,4524],{},"β",[1820,4526,1835],{"fence":1845},[1813,4528,4529,4531,4533,4535],{},[1820,4530,1986],{},[2132,4532,2134],{},[1816,4534,1994],{"mathvariant":1993},[1816,4536,2350],{},[1820,4538,2542],{"separator":1845},[1837,4540,4541],{"encoding":1839},"P(Y>y)=\\left(1+\\xi\\frac{y}{\\beta}\\right)^{-1\u002F\\xi},",[1797,4543,4545,4569,4591],{"className":4544,"ariaHidden":1845},[1844],[1797,4546,4548,4551,4554,4557,4560,4563,4566],{"className":4547},[1849],[1797,4549],{"className":4550,"style":1854},[1853],[1797,4552,1818],{"className":4553,"style":1860},[1858,1859],[1797,4555,1823],{"className":4556},[1864],[1797,4558,4367],{"className":4559,"style":2177},[1858,1859],[1797,4561],{"className":4562,"style":1873},[1872],[1797,4564,1829],{"className":4565},[1877],[1797,4567],{"className":4568,"style":1873},[1872],[1797,4570,4572,4575,4579,4582,4585,4588],{"className":4571},[1849],[1797,4573],{"className":4574,"style":1854},[1853],[1797,4576,4501],{"className":4577,"style":4578},[1858,1859],"margin-right:0.0359em;",[1797,4580,1835],{"className":4581},[1893],[1797,4583],{"className":4584,"style":1873},[1872],[1797,4586,2577],{"className":4587},[1877],[1797,4589],{"className":4590,"style":1873},[1872],[1797,4592,4594,4598,4730,4733],{"className":4593},[1849],[1797,4595],{"className":4596,"style":4597},[1853],"height:2.6779em;vertical-align:-0.95em;",[1797,4599,4601,4695],{"className":4600},[3270],[1797,4602,4604,4610,4613,4616,4619,4622,4625,4689],{"className":4603},[3270],[1797,4605,4607],{"className":4606,"style":3278},[1864,3277],[1797,4608,1823],{"className":4609},[3282,2080],[1797,4611,2134],{"className":4612},[1858],[1797,4614],{"className":4615,"style":2177},[1872],[1797,4617,2137],{"className":4618},[2181],[1797,4620],{"className":4621,"style":2177},[1872],[1797,4623,2350],{"className":4624,"style":2367},[1858,1859],[1797,4626,4628,4631,4686],{"className":4627},[1858],[1797,4629],{"className":4630},[1864,2825],[1797,4632,4634],{"className":4633},[2619],[1797,4635,4637,4677],{"className":4636},[2057,2832],[1797,4638,4640,4674],{"className":4639},[2061],[1797,4641,4643,4655,4663],{"className":4642,"style":3314},[2065],[1797,4644,4645,4648],{"style":2842},[1797,4646],{"className":4647,"style":2681},[2073],[1797,4649,4651],{"className":4650},[1858],[1797,4652,4524],{"className":4653,"style":4654},[1858,1859],"margin-right:0.0528em;",[1797,4656,4657,4660],{"style":2854},[1797,4658],{"className":4659,"style":2681},[2073],[1797,4661],{"className":4662,"style":2862},[2861],[1797,4664,4665,4668],{"style":2865},[1797,4666],{"className":4667,"style":2681},[2073],[1797,4669,4671],{"className":4670},[1858],[1797,4672,4501],{"className":4673,"style":4578},[1858,1859],[1797,4675,2879],{"className":4676},[2878],[1797,4678,4680],{"className":4679},[2061],[1797,4681,4684],{"className":4682,"style":4683},[2065],"height:0.8804em;",[1797,4685],{},[1797,4687],{"className":4688},[1893,2825],[1797,4690,4692],{"className":4691,"style":3278},[1893,3277],[1797,4693,1835],{"className":4694},[3282,2080],[1797,4696,4698],{"className":4697},[2053],[1797,4699,4701],{"className":4700},[2057],[1797,4702,4704],{"className":4703},[2061],[1797,4705,4708],{"className":4706,"style":4707},[2065],"height:1.7279em;",[1797,4709,4711,4714],{"style":4710},"top:-3.9029em;margin-right:0.05em;",[1797,4712],{"className":4713,"style":2074},[2073],[1797,4715,4717],{"className":4716},[2078,2079,2080,2081],[1797,4718,4720,4723,4727],{"className":4719},[1858,2081],[1797,4721,1986],{"className":4722},[1858,2081],[1797,4724,4726],{"className":4725},[1858,2081],"1\u002F",[1797,4728,2350],{"className":4729,"style":2367},[1858,1859,2081],[1797,4731],{"className":4732,"style":2790},[1872],[1797,4734,2542],{"className":4735},[2782],[1793,4737,4738,4739,4828,4829,2490,4880,2647],{},"on the support where ",[1797,4740,4742,4770],{"className":4741},[1800],[1797,4743,4745],{"className":4744},[1804],[1806,4746,4747],{"xmlns":1808},[1810,4748,4749,4767],{},[1813,4750,4751,4753,4755,4757,4759,4761,4763,4765],{},[2132,4752,2134],{},[1820,4754,2137],{},[1816,4756,2350],{},[1816,4758,4501],{},[1816,4760,1994],{"mathvariant":1993},[1816,4762,4524],{},[1820,4764,1829],{},[2132,4766,2455],{},[1837,4768,4769],{"encoding":1839},"1+\\xi y\u002F\\beta>0",[1797,4771,4773,4792,4819],{"className":4772,"ariaHidden":1845},[1844],[1797,4774,4776,4780,4783,4786,4789],{"className":4775},[1849],[1797,4777],{"className":4778,"style":4779},[1853],"height:0.7278em;vertical-align:-0.0833em;",[1797,4781,2134],{"className":4782},[1858],[1797,4784],{"className":4785,"style":2177},[1872],[1797,4787,2137],{"className":4788},[2181],[1797,4790],{"className":4791,"style":2177},[1872],[1797,4793,4795,4798,4801,4804,4807,4810,4813,4816],{"className":4794},[1849],[1797,4796],{"className":4797,"style":1854},[1853],[1797,4799,2350],{"className":4800,"style":2367},[1858,1859],[1797,4802,4501],{"className":4803,"style":4578},[1858,1859],[1797,4805,1994],{"className":4806},[1858],[1797,4808,4524],{"className":4809,"style":4654},[1858,1859],[1797,4811],{"className":4812,"style":1873},[1872],[1797,4814,1829],{"className":4815},[1877],[1797,4817],{"className":4818,"style":1873},[1872],[1797,4820,4822,4825],{"className":4821},[1849],[1797,4823],{"className":4824,"style":2420},[1853],[1797,4826,2455],{"className":4827},[1858],", with scale ",[1797,4830,4832,4850],{"className":4831},[1800],[1797,4833,4835],{"className":4834},[1804],[1806,4836,4837],{"xmlns":1808},[1810,4838,4839,4847],{},[1813,4840,4841,4843,4845],{},[1816,4842,4524],{},[1820,4844,1829],{},[2132,4846,2455],{},[1837,4848,4849],{"encoding":1839},"\\beta>0",[1797,4851,4853,4871],{"className":4852,"ariaHidden":1845},[1844],[1797,4854,4856,4859,4862,4865,4868],{"className":4855},[1849],[1797,4857],{"className":4858,"style":2363},[1853],[1797,4860,4524],{"className":4861,"style":4654},[1858,1859],[1797,4863],{"className":4864,"style":1873},[1872],[1797,4866,1829],{"className":4867},[1877],[1797,4869],{"className":4870,"style":1873},[1872],[1797,4872,4874,4877],{"className":4873},[1849],[1797,4875],{"className":4876,"style":2420},[1853],[1797,4878,2455],{"className":4879},[1858],[1797,4881,4883,4896],{"className":4882},[1800],[1797,4884,4886],{"className":4885},[1804],[1806,4887,4888],{"xmlns":1808},[1810,4889,4890,4894],{},[1813,4891,4892],{},[1816,4893,2350],{},[1837,4895,2353],{"encoding":1839},[1797,4897,4899],{"className":4898,"ariaHidden":1845},[1844],[1797,4900,4902,4905],{"className":4901},[1849],[1797,4903],{"className":4904,"style":2363},[1853],[1797,4906,2350],{"className":4907,"style":2367},[1858,1859],[1931,4909,4910,4924],{},[1934,4911,4912],{},[1937,4913,4914,4918,4921],{},[1940,4915,4917],{"align":4916},"right","Shape",[1940,4919,4920],{},"Tail",[1940,4922,4923],{},"Consequence",[1953,4925,4926,4987,5048],{},[1937,4927,4928,4981,4984],{},[1958,4929,4930],{"align":4916},[1797,4931,4933,4951],{"className":4932},[1800],[1797,4934,4936],{"className":4935},[1804],[1806,4937,4938],{"xmlns":1808},[1810,4939,4940,4948],{},[1813,4941,4942,4944,4946],{},[1816,4943,2350],{},[1820,4945,2288],{},[2132,4947,2455],{},[1837,4949,4950],{"encoding":1839},"\\xi\u003C0",[1797,4952,4954,4972],{"className":4953,"ariaHidden":1845},[1844],[1797,4955,4957,4960,4963,4966,4969],{"className":4956},[1849],[1797,4958],{"className":4959,"style":2363},[1853],[1797,4961,2350],{"className":4962,"style":2367},[1858,1859],[1797,4964],{"className":4965,"style":1873},[1872],[1797,4967,2288],{"className":4968},[1877],[1797,4970],{"className":4971,"style":1873},[1872],[1797,4973,4975,4978],{"className":4974},[1849],[1797,4976],{"className":4977,"style":2420},[1853],[1797,4979,2455],{"className":4980},[1858],[1958,4982,4983],{},"finite upper endpoint",[1958,4985,4986],{},"useful only when a genuine cap exists",[1937,4988,4989,5042,5045],{},[1958,4990,4991],{"align":4916},[1797,4992,4994,5012],{"className":4993},[1800],[1797,4995,4997],{"className":4996},[1804],[1806,4998,4999],{"xmlns":1808},[1810,5000,5001,5009],{},[1813,5002,5003,5005,5007],{},[1816,5004,2350],{},[1820,5006,2577],{},[2132,5008,2455],{},[1837,5010,5011],{"encoding":1839},"\\xi=0",[1797,5013,5015,5033],{"className":5014,"ariaHidden":1845},[1844],[1797,5016,5018,5021,5024,5027,5030],{"className":5017},[1849],[1797,5019],{"className":5020,"style":2363},[1853],[1797,5022,2350],{"className":5023,"style":2367},[1858,1859],[1797,5025],{"className":5026,"style":1873},[1872],[1797,5028,2577],{"className":5029},[1877],[1797,5031],{"className":5032,"style":1873},[1872],[1797,5034,5036,5039],{"className":5035},[1849],[1797,5037],{"className":5038,"style":2420},[1853],[1797,5040,2455],{"className":5041},[1858],[1958,5043,5044],{},"Exponential limit",[1958,5046,5047],{},"light tail",[1937,5049,5050,5103,5106],{},[1958,5051,5052],{"align":4916},[1797,5053,5055,5073],{"className":5054},[1800],[1797,5056,5058],{"className":5057},[1804],[1806,5059,5060],{"xmlns":1808},[1810,5061,5062,5070],{},[1813,5063,5064,5066,5068],{},[1816,5065,2350],{},[1820,5067,1829],{},[2132,5069,2455],{},[1837,5071,5072],{"encoding":1839},"\\xi>0",[1797,5074,5076,5094],{"className":5075,"ariaHidden":1845},[1844],[1797,5077,5079,5082,5085,5088,5091],{"className":5078},[1849],[1797,5080],{"className":5081,"style":2363},[1853],[1797,5083,2350],{"className":5084,"style":2367},[1858,1859],[1797,5086],{"className":5087,"style":1873},[1872],[1797,5089,1829],{"className":5090},[1877],[1797,5092],{"className":5093,"style":1873},[1872],[1797,5095,5097,5100],{"className":5096},[1849],[1797,5098],{"className":5099,"style":2420},[1853],[1797,5101,2455],{"className":5102},[1858],[1958,5104,5105],{},"Pareto-type heavy tail",[1958,5107,5108,5109,5159,5160],{},"mean finite only if ",[1797,5110,5112,5129],{"className":5111},[1800],[1797,5113,5115],{"className":5114},[1804],[1806,5116,5117],{"xmlns":1808},[1810,5118,5119,5127],{},[1813,5120,5121,5123,5125],{},[1816,5122,2350],{},[1820,5124,2288],{},[2132,5126,2134],{},[1837,5128,2392],{"encoding":1839},[1797,5130,5132,5150],{"className":5131,"ariaHidden":1845},[1844],[1797,5133,5135,5138,5141,5144,5147],{"className":5134},[1849],[1797,5136],{"className":5137,"style":2363},[1853],[1797,5139,2350],{"className":5140,"style":2367},[1858,1859],[1797,5142],{"className":5143,"style":1873},[1872],[1797,5145,2288],{"className":5146},[1877],[1797,5148],{"className":5149,"style":1873},[1872],[1797,5151,5153,5156],{"className":5152},[1849],[1797,5154],{"className":5155,"style":2420},[1853],[1797,5157,2134],{"className":5158},[1858],"; variance only if ",[1797,5161,5163,5185],{"className":5162},[1800],[1797,5164,5166],{"className":5165},[1804],[1806,5167,5168],{"xmlns":1808},[1810,5169,5170,5182],{},[1813,5171,5172,5174,5176,5178,5180],{},[1816,5173,2350],{},[1820,5175,2288],{},[2132,5177,2134],{},[1816,5179,1994],{"mathvariant":1993},[2132,5181,3711],{},[1837,5183,5184],{"encoding":1839},"\\xi\u003C1\u002F2",[1797,5186,5188,5206],{"className":5187,"ariaHidden":1845},[1844],[1797,5189,5191,5194,5197,5200,5203],{"className":5190},[1849],[1797,5192],{"className":5193,"style":2363},[1853],[1797,5195,2350],{"className":5196,"style":2367},[1858,1859],[1797,5198],{"className":5199,"style":1873},[1872],[1797,5201,2288],{"className":5202},[1877],[1797,5204],{"className":5205,"style":1873},[1872],[1797,5207,5209,5212],{"className":5208},[1849],[1797,5210],{"className":5211,"style":1854},[1853],[1797,5213,5215],{"className":5214},[1858],"1\u002F2",[1793,5217,5218,5219,2542],{},"For ",[1797,5220,5222,5240],{"className":5221},[1800],[1797,5223,5225],{"className":5224},[1804],[1806,5226,5227],{"xmlns":1808},[1810,5228,5229,5237],{},[1813,5230,5231,5233,5235],{},[1816,5232,1832],{},[1820,5234,1829],{},[1816,5236,4337],{},[1837,5238,5239],{"encoding":1839},"x>u",[1797,5241,5243,5261],{"className":5242,"ariaHidden":1845},[1844],[1797,5244,5246,5249,5252,5255,5258],{"className":5245},[1849],[1797,5247],{"className":5248,"style":2303},[1853],[1797,5250,1832],{"className":5251},[1858,1859],[1797,5253],{"className":5254,"style":1873},[1872],[1797,5256,1829],{"className":5257},[1877],[1797,5259],{"className":5260,"style":1873},[1872],[1797,5262,5264,5267],{"className":5263},[1849],[1797,5265],{"className":5266,"style":1920},[1853],[1797,5268,4337],{"className":5269},[1858,1859],[1797,5271,5273],{"className":5272},[2546],[1797,5274,5276,5344],{"className":5275},[1800],[1797,5277,5279],{"className":5278},[1804],[1806,5280,5281],{"xmlns":1808,"display":2555},[1810,5282,5283,5341],{},[1813,5284,5285,5287,5289,5291,5293,5295,5297,5299,5301,5303,5305,5307,5309,5311,5313,5315,5317,5319,5321,5323,5325,5327,5329,5331,5333,5335,5337,5339],{},[1816,5286,1818],{},[1820,5288,1823],{"stretchy":1822},[1816,5290,1826],{},[1820,5292,1829],{},[1816,5294,1832],{},[1820,5296,1835],{"stretchy":1822},[1820,5298,2577],{},[1816,5300,1818],{},[1820,5302,1823],{"stretchy":1822},[1816,5304,1826],{},[1820,5306,1829],{},[1816,5308,4337],{},[1820,5310,1835],{"stretchy":1822},[1816,5312,1818],{},[1820,5314,1823],{"stretchy":1822},[1816,5316,1826],{},[1820,5318,1986],{},[1816,5320,4337],{},[1820,5322,1829],{},[1816,5324,1832],{},[1820,5326,1986],{},[1816,5328,4337],{},[1820,5330,4378],{},[1816,5332,1826],{},[1820,5334,1829],{},[1816,5336,4337],{},[1820,5338,1835],{"stretchy":1822},[1816,5340,2647],{"mathvariant":1993},[1837,5342,5343],{"encoding":1839},"P(X>x)=P(X>u)P(X-u>x-u\\mid X>u).",[1797,5345,5347,5371,5392,5416,5446,5464,5483,5501,5519],{"className":5346,"ariaHidden":1845},[1844],[1797,5348,5350,5353,5356,5359,5362,5365,5368],{"className":5349},[1849],[1797,5351],{"className":5352,"style":1854},[1853],[1797,5354,1818],{"className":5355,"style":1860},[1858,1859],[1797,5357,1823],{"className":5358},[1864],[1797,5360,1826],{"className":5361,"style":1868},[1858,1859],[1797,5363],{"className":5364,"style":1873},[1872],[1797,5366,1829],{"className":5367},[1877],[1797,5369],{"className":5370,"style":1873},[1872],[1797,5372,5374,5377,5380,5383,5386,5389],{"className":5373},[1849],[1797,5375],{"className":5376,"style":1854},[1853],[1797,5378,1832],{"className":5379},[1858,1859],[1797,5381,1835],{"className":5382},[1893],[1797,5384],{"className":5385,"style":1873},[1872],[1797,5387,2577],{"className":5388},[1877],[1797,5390],{"className":5391,"style":1873},[1872],[1797,5393,5395,5398,5401,5404,5407,5410,5413],{"className":5394},[1849],[1797,5396],{"className":5397,"style":1854},[1853],[1797,5399,1818],{"className":5400,"style":1860},[1858,1859],[1797,5402,1823],{"className":5403},[1864],[1797,5405,1826],{"className":5406,"style":1868},[1858,1859],[1797,5408],{"className":5409,"style":1873},[1872],[1797,5411,1829],{"className":5412},[1877],[1797,5414],{"className":5415,"style":1873},[1872],[1797,5417,5419,5422,5425,5428,5431,5434,5437,5440,5443],{"className":5418},[1849],[1797,5420],{"className":5421,"style":1854},[1853],[1797,5423,4337],{"className":5424},[1858,1859],[1797,5426,1835],{"className":5427},[1893],[1797,5429,1818],{"className":5430,"style":1860},[1858,1859],[1797,5432,1823],{"className":5433},[1864],[1797,5435,1826],{"className":5436,"style":1868},[1858,1859],[1797,5438],{"className":5439,"style":2177},[1872],[1797,5441,1986],{"className":5442},[2181],[1797,5444],{"className":5445,"style":2177},[1872],[1797,5447,5449,5452,5455,5458,5461],{"className":5448},[1849],[1797,5450],{"className":5451,"style":2303},[1853],[1797,5453,4337],{"className":5454},[1858,1859],[1797,5456],{"className":5457,"style":1873},[1872],[1797,5459,1829],{"className":5460},[1877],[1797,5462],{"className":5463,"style":1873},[1872],[1797,5465,5467,5471,5474,5477,5480],{"className":5466},[1849],[1797,5468],{"className":5469,"style":5470},[1853],"height:0.6667em;vertical-align:-0.0833em;",[1797,5472,1832],{"className":5473},[1858,1859],[1797,5475],{"className":5476,"style":2177},[1872],[1797,5478,1986],{"className":5479},[2181],[1797,5481],{"className":5482,"style":2177},[1872],[1797,5484,5486,5489,5492,5495,5498],{"className":5485},[1849],[1797,5487],{"className":5488,"style":1854},[1853],[1797,5490,4337],{"className":5491},[1858,1859],[1797,5493],{"className":5494,"style":1873},[1872],[1797,5496,4378],{"className":5497},[1877],[1797,5499],{"className":5500,"style":1873},[1872],[1797,5502,5504,5507,5510,5513,5516],{"className":5503},[1849],[1797,5505],{"className":5506,"style":4453},[1853],[1797,5508,1826],{"className":5509,"style":1868},[1858,1859],[1797,5511],{"className":5512,"style":1873},[1872],[1797,5514,1829],{"className":5515},[1877],[1797,5517],{"className":5518,"style":1873},[1872],[1797,5520,5522,5525,5528,5531],{"className":5521},[1849],[1797,5523],{"className":5524,"style":1854},[1853],[1797,5526,4337],{"className":5527},[1858,1859],[1797,5529,1835],{"className":5530},[1893],[1797,5532,2647],{"className":5533},[1858],[1793,5535,5536,5537,5603],{},"Both the exceedance rate ",[1797,5538,5540,5564],{"className":5539},[1800],[1797,5541,5543],{"className":5542},[1804],[1806,5544,5545],{"xmlns":1808},[1810,5546,5547,5561],{},[1813,5548,5549,5551,5553,5555,5557,5559],{},[1816,5550,1818],{},[1820,5552,1823],{"stretchy":1822},[1816,5554,1826],{},[1820,5556,1829],{},[1816,5558,4337],{},[1820,5560,1835],{"stretchy":1822},[1837,5562,5563],{"encoding":1839},"P(X>u)",[1797,5565,5567,5591],{"className":5566,"ariaHidden":1845},[1844],[1797,5568,5570,5573,5576,5579,5582,5585,5588],{"className":5569},[1849],[1797,5571],{"className":5572,"style":1854},[1853],[1797,5574,1818],{"className":5575,"style":1860},[1858,1859],[1797,5577,1823],{"className":5578},[1864],[1797,5580,1826],{"className":5581,"style":1868},[1858,1859],[1797,5583],{"className":5584,"style":1873},[1872],[1797,5586,1829],{"className":5587},[1877],[1797,5589],{"className":5590,"style":1873},[1872],[1797,5592,5594,5597,5600],{"className":5593},[1849],[1797,5595],{"className":5596,"style":1854},[1853],[1797,5598,4337],{"className":5599},[1858,1859],[1797,5601,1835],{"className":5602},[1893]," and the conditional excess model are needed to price a layer.",[1926,5605,5607],{"id":5606},"_5-threshold-choice-is-a-biasvariance-decision","5. Threshold choice is a bias–variance decision",[2946,5609,5610,5613],{},[2949,5611,5612],{},"Too low: the asymptotic GPD approximation may be biased.",[2949,5614,5615],{},"Too high: very few exceedances create unstable shape estimates.",[1793,5617,5618],{},"Use several views together:",[5620,5621,5622,5625,5628,5631,5634],"ol",{},[2949,5623,5624],{},"mean residual life plot;",[2949,5626,5627],{},"parameter stability over candidate thresholds;",[2949,5629,5630],{},"QQ\u002FPP diagnostics for exceedances;",[2949,5632,5633],{},"number and independence of exceedances;",[2949,5635,5636],{},"sensitivity of the actual target—quantile or layer price.",[1793,5638,5639],{},"A plot does not “select” the threshold automatically. Report a plausible range and show consequence sensitivity.",[1926,5641,5643],{"id":5642},"_6-worked-layer-calculation","6. Worked layer calculation",[1793,5645,5646,5647,5751,5752,5804,5805,5869,5870,2647],{},"Suppose ",[1797,5648,5650,5683],{"className":5649},[1800],[1797,5651,5653],{"className":5652},[1804],[1806,5654,5655],{"xmlns":1808},[1810,5656,5657,5680],{},[1813,5658,5659,5661,5663,5665,5667,5670,5673,5675,5677],{},[1816,5660,1818],{},[1820,5662,1823],{"stretchy":1822},[1816,5664,1826],{},[1820,5666,1829],{},[1816,5668,5669],{"mathvariant":1993},"£",[2132,5671,5672],{},"100,000",[1820,5674,1835],{"stretchy":1822},[1820,5676,2577],{},[2132,5678,5679],{},"0.02",[1837,5681,5682],{"encoding":1839},"P(X>£100{,}000)=0.02",[1797,5684,5686,5710,5742],{"className":5685,"ariaHidden":1845},[1844],[1797,5687,5689,5692,5695,5698,5701,5704,5707],{"className":5688},[1849],[1797,5690],{"className":5691,"style":1854},[1853],[1797,5693,1818],{"className":5694,"style":1860},[1858,1859],[1797,5696,1823],{"className":5697},[1864],[1797,5699,1826],{"className":5700,"style":1868},[1858,1859],[1797,5702],{"className":5703,"style":1873},[1872],[1797,5705,1829],{"className":5706},[1877],[1797,5708],{"className":5709,"style":1873},[1872],[1797,5711,5713,5716,5720,5726,5730,5733,5736,5739],{"className":5712},[1849],[1797,5714],{"className":5715,"style":1854},[1853],[1797,5717,5719],{"className":5718},[1858],"£100",[1797,5721,5723],{"className":5722},[1858],[1797,5724,2542],{"className":5725},[2782],[1797,5727,5729],{"className":5728},[1858],"000",[1797,5731,1835],{"className":5732},[1893],[1797,5734],{"className":5735,"style":1873},[1872],[1797,5737,2577],{"className":5738},[1877],[1797,5740],{"className":5741,"style":1873},[1872],[1797,5743,5745,5748],{"className":5744},[1849],[1797,5746],{"className":5747,"style":2420},[1853],[1797,5749,5679],{"className":5750},[1858],". Above £100,000, excess follows a GPD with ",[1797,5753,5755,5774],{"className":5754},[1800],[1797,5756,5758],{"className":5757},[1804],[1806,5759,5760],{"xmlns":1808},[1810,5761,5762,5771],{},[1813,5763,5764,5766,5768],{},[1816,5765,2350],{},[1820,5767,2577],{},[2132,5769,5770],{},"0.25",[1837,5772,5773],{"encoding":1839},"\\xi=0.25",[1797,5775,5777,5795],{"className":5776,"ariaHidden":1845},[1844],[1797,5778,5780,5783,5786,5789,5792],{"className":5779},[1849],[1797,5781],{"className":5782,"style":2363},[1853],[1797,5784,2350],{"className":5785,"style":2367},[1858,1859],[1797,5787],{"className":5788,"style":1873},[1872],[1797,5790,2577],{"className":5791},[1877],[1797,5793],{"className":5794,"style":1873},[1872],[1797,5796,5798,5801],{"className":5797},[1849],[1797,5799],{"className":5800,"style":2420},[1853],[1797,5802,5770],{"className":5803},[1858]," and ",[1797,5806,5808,5829],{"className":5807},[1800],[1797,5809,5811],{"className":5810},[1804],[1806,5812,5813],{"xmlns":1808},[1810,5814,5815,5826],{},[1813,5816,5817,5819,5821,5823],{},[1816,5818,4524],{},[1820,5820,2577],{},[1816,5822,5669],{"mathvariant":1993},[2132,5824,5825],{},"50,000",[1837,5827,5828],{"encoding":1839},"\\beta=£50{,}000",[1797,5830,5832,5850],{"className":5831,"ariaHidden":1845},[1844],[1797,5833,5835,5838,5841,5844,5847],{"className":5834},[1849],[1797,5836],{"className":5837,"style":2363},[1853],[1797,5839,4524],{"className":5840,"style":4654},[1858,1859],[1797,5842],{"className":5843,"style":1873},[1872],[1797,5845,2577],{"className":5846},[1877],[1797,5848],{"className":5849,"style":1873},[1872],[1797,5851,5853,5856,5860,5866],{"className":5852},[1849],[1797,5854],{"className":5855,"style":2363},[1853],[1797,5857,5859],{"className":5858},[1858],"£50",[1797,5861,5863],{"className":5862},[1858],[1797,5864,2542],{"className":5865},[2782],[1797,5867,5729],{"className":5868},[1858],". Estimate ",[1797,5871,5873,5900],{"className":5872},[1800],[1797,5874,5876],{"className":5875},[1804],[1806,5877,5878],{"xmlns":1808},[1810,5879,5880,5897],{},[1813,5881,5882,5884,5886,5888,5890,5892,5895],{},[1816,5883,1818],{},[1820,5885,1823],{"stretchy":1822},[1816,5887,1826],{},[1820,5889,1829],{},[1816,5891,5669],{"mathvariant":1993},[2132,5893,5894],{},"300,000",[1820,5896,1835],{"stretchy":1822},[1837,5898,5899],{"encoding":1839},"P(X>£300{,}000)",[1797,5901,5903,5927],{"className":5902,"ariaHidden":1845},[1844],[1797,5904,5906,5909,5912,5915,5918,5921,5924],{"className":5905},[1849],[1797,5907],{"className":5908,"style":1854},[1853],[1797,5910,1818],{"className":5911,"style":1860},[1858,1859],[1797,5913,1823],{"className":5914},[1864],[1797,5916,1826],{"className":5917,"style":1868},[1858,1859],[1797,5919],{"className":5920,"style":1873},[1872],[1797,5922,1829],{"className":5923},[1877],[1797,5925],{"className":5926,"style":1873},[1872],[1797,5928,5930,5933,5937,5943,5946],{"className":5929},[1849],[1797,5931],{"className":5932,"style":1854},[1853],[1797,5934,5936],{"className":5935},[1858],"£300",[1797,5938,5940],{"className":5939},[1858],[1797,5941,2542],{"className":5942},[2782],[1797,5944,5729],{"className":5945},[1858],[1797,5947,1835],{"className":5948},[1893],[1793,5950,5951,5952,6017],{},"Here ",[1797,5953,5955,5976],{"className":5954},[1800],[1797,5956,5958],{"className":5957},[1804],[1806,5959,5960],{"xmlns":1808},[1810,5961,5962,5973],{},[1813,5963,5964,5966,5968,5970],{},[1816,5965,4501],{},[1820,5967,2577],{},[1816,5969,5669],{"mathvariant":1993},[2132,5971,5972],{},"200,000",[1837,5974,5975],{"encoding":1839},"y=£200{,}000",[1797,5977,5979,5998],{"className":5978,"ariaHidden":1845},[1844],[1797,5980,5982,5986,5989,5992,5995],{"className":5981},[1849],[1797,5983],{"className":5984,"style":5985},[1853],"height:0.625em;vertical-align:-0.1944em;",[1797,5987,4501],{"className":5988,"style":4578},[1858,1859],[1797,5990],{"className":5991,"style":1873},[1872],[1797,5993,2577],{"className":5994},[1877],[1797,5996],{"className":5997,"style":1873},[1872],[1797,5999,6001,6004,6008,6014],{"className":6000},[1849],[1797,6002],{"className":6003,"style":2363},[1853],[1797,6005,6007],{"className":6006},[1858],"£200",[1797,6009,6011],{"className":6010},[1858],[1797,6012,2542],{"className":6013},[2782],[1797,6015,5729],{"className":6016},[1858],":",[1797,6019,6021],{"className":6020},[2546],[1797,6022,6024,6094],{"className":6023},[1800],[1797,6025,6027],{"className":6026},[1804],[1806,6028,6029],{"xmlns":1808,"display":2555},[1810,6030,6031,6091],{},[1813,6032,6033,6035,6037,6039,6041,6043,6045,6047,6074,6076,6086,6088],{},[1816,6034,1818],{},[1820,6036,1823],{"stretchy":1822},[1816,6038,4367],{},[1820,6040,1829],{},[2132,6042,5972],{},[1820,6044,1835],{"stretchy":1822},[1820,6046,2577],{},[1999,6048,6049,6067],{},[1813,6050,6051,6053,6055,6057,6059,6065],{},[1820,6052,1823],{"fence":1845},[2132,6054,2134],{},[1820,6056,2137],{},[2132,6058,5770],{},[2619,6060,6061,6063],{},[2132,6062,5972],{},[2132,6064,5825],{},[1820,6066,1835],{"fence":1845},[1813,6068,6069,6071],{},[1820,6070,1986],{},[2132,6072,6073],{},"4",[1820,6075,2577],{},[1999,6077,6078,6080],{},[2132,6079,3711],{},[1813,6081,6082,6084],{},[1820,6083,1986],{},[2132,6085,6073],{},[1820,6087,2577],{},[2132,6089,6090],{},"0.0625.",[1837,6092,6093],{"encoding":1839},"P(Y>200{,}000)\n=\\left(1+0.25\\frac{200{,}000}{50{,}000}\\right)^{-4}\n=2^{-4}=0.0625.",[1797,6095,6097,6121,6152,6313,6364],{"className":6096,"ariaHidden":1845},[1844],[1797,6098,6100,6103,6106,6109,6112,6115,6118],{"className":6099},[1849],[1797,6101],{"className":6102,"style":1854},[1853],[1797,6104,1818],{"className":6105,"style":1860},[1858,1859],[1797,6107,1823],{"className":6108},[1864],[1797,6110,4367],{"className":6111,"style":2177},[1858,1859],[1797,6113],{"className":6114,"style":1873},[1872],[1797,6116,1829],{"className":6117},[1877],[1797,6119],{"className":6120,"style":1873},[1872],[1797,6122,6124,6127,6131,6137,6140,6143,6146,6149],{"className":6123},[1849],[1797,6125],{"className":6126,"style":1854},[1853],[1797,6128,6130],{"className":6129},[1858],"200",[1797,6132,6134],{"className":6133},[1858],[1797,6135,2542],{"className":6136},[2782],[1797,6138,5729],{"className":6139},[1858],[1797,6141,1835],{"className":6142},[1893],[1797,6144],{"className":6145,"style":1873},[1872],[1797,6147,2577],{"className":6148},[1877],[1797,6150],{"className":6151,"style":1873},[1872],[1797,6153,6155,6159,6304,6307,6310],{"className":6154},[1849],[1797,6156],{"className":6157,"style":6158},[1853],"height:2.604em;vertical-align:-0.95em;",[1797,6160,6162,6274],{"className":6161},[3270],[1797,6163,6165,6171,6174,6177,6180,6183,6186,6268],{"className":6164},[3270],[1797,6166,6168],{"className":6167,"style":3278},[1864,3277],[1797,6169,1823],{"className":6170},[3282,2080],[1797,6172,2134],{"className":6173},[1858],[1797,6175],{"className":6176,"style":2177},[1872],[1797,6178,2137],{"className":6179},[2181],[1797,6181],{"className":6182,"style":2177},[1872],[1797,6184,5770],{"className":6185},[1858],[1797,6187,6189,6192,6265],{"className":6188},[1858],[1797,6190],{"className":6191},[1864,2825],[1797,6193,6195],{"className":6194},[2619],[1797,6196,6198,6257],{"className":6197},[2057,2832],[1797,6199,6201,6254],{"className":6200},[2061],[1797,6202,6205,6226,6234],{"className":6203,"style":6204},[2065],"height:1.3214em;",[1797,6206,6207,6210],{"style":2842},[1797,6208],{"className":6209,"style":2681},[2073],[1797,6211,6213,6217,6223],{"className":6212},[1858],[1797,6214,6216],{"className":6215},[1858],"50",[1797,6218,6220],{"className":6219},[1858],[1797,6221,2542],{"className":6222},[2782],[1797,6224,5729],{"className":6225},[1858],[1797,6227,6228,6231],{"style":2854},[1797,6229],{"className":6230,"style":2681},[2073],[1797,6232],{"className":6233,"style":2862},[2861],[1797,6235,6236,6239],{"style":2865},[1797,6237],{"className":6238,"style":2681},[2073],[1797,6240,6242,6245,6251],{"className":6241},[1858],[1797,6243,6130],{"className":6244},[1858],[1797,6246,6248],{"className":6247},[1858],[1797,6249,2542],{"className":6250},[2782],[1797,6252,5729],{"className":6253},[1858],[1797,6255,2879],{"className":6256},[2878],[1797,6258,6260],{"className":6259},[2061],[1797,6261,6263],{"className":6262,"style":4683},[2065],[1797,6264],{},[1797,6266],{"className":6267},[1893,2825],[1797,6269,6271],{"className":6270,"style":3278},[1893,3277],[1797,6272,1835],{"className":6273},[3282,2080],[1797,6275,6277],{"className":6276},[2053],[1797,6278,6280],{"className":6279},[2057],[1797,6281,6283],{"className":6282},[2061],[1797,6284,6287],{"className":6285,"style":6286},[2065],"height:1.654em;",[1797,6288,6289,6292],{"style":4710},[1797,6290],{"className":6291,"style":2074},[2073],[1797,6293,6295],{"className":6294},[2078,2079,2080,2081],[1797,6296,6298,6301],{"className":6297},[1858,2081],[1797,6299,1986],{"className":6300},[1858,2081],[1797,6302,6073],{"className":6303},[1858,2081],[1797,6305],{"className":6306,"style":1873},[1872],[1797,6308,2577],{"className":6309},[1877],[1797,6311],{"className":6312,"style":1873},[1872],[1797,6314,6316,6320,6355,6358,6361],{"className":6315},[1849],[1797,6317],{"className":6318,"style":6319},[1853],"height:0.8641em;",[1797,6321,6323,6326],{"className":6322},[1858],[1797,6324,3711],{"className":6325},[1858],[1797,6327,6329],{"className":6328},[2053],[1797,6330,6332],{"className":6331},[2057],[1797,6333,6335],{"className":6334},[2061],[1797,6336,6338],{"className":6337,"style":6319},[2065],[1797,6339,6340,6343],{"style":2766},[1797,6341],{"className":6342,"style":2074},[2073],[1797,6344,6346],{"className":6345},[2078,2079,2080,2081],[1797,6347,6349,6352],{"className":6348},[1858,2081],[1797,6350,1986],{"className":6351},[1858,2081],[1797,6353,6073],{"className":6354},[1858,2081],[1797,6356],{"className":6357,"style":1873},[1872],[1797,6359,2577],{"className":6360},[1877],[1797,6362],{"className":6363,"style":1873},[1872],[1797,6365,6367,6370],{"className":6366},[1849],[1797,6368],{"className":6369,"style":2420},[1853],[1797,6371,6090],{"className":6372},[1858],[1793,6374,6375],{},"Therefore",[1797,6377,6379],{"className":6378},[2546],[1797,6380,6382,6421],{"className":6381},[1800],[1797,6383,6385],{"className":6384},[1804],[1806,6386,6387],{"xmlns":1808,"display":2555},[1810,6388,6389,6418],{},[1813,6390,6391,6393,6395,6397,6399,6401,6403,6405,6407,6410,6413,6415],{},[1816,6392,1818],{},[1820,6394,1823],{"stretchy":1822},[1816,6396,1826],{},[1820,6398,1829],{},[2132,6400,5894],{},[1820,6402,1835],{"stretchy":1822},[1820,6404,2577],{},[2132,6406,5679],{},[1820,6408,6409],{},"×",[2132,6411,6412],{},"0.0625",[1820,6414,2577],{},[2132,6416,6417],{},"0.00125.",[1837,6419,6420],{"encoding":1839},"P(X>300{,}000)=0.02\\times0.0625=0.00125.",[1797,6422,6424,6448,6479,6497,6515],{"className":6423,"ariaHidden":1845},[1844],[1797,6425,6427,6430,6433,6436,6439,6442,6445],{"className":6426},[1849],[1797,6428],{"className":6429,"style":1854},[1853],[1797,6431,1818],{"className":6432,"style":1860},[1858,1859],[1797,6434,1823],{"className":6435},[1864],[1797,6437,1826],{"className":6438,"style":1868},[1858,1859],[1797,6440],{"className":6441,"style":1873},[1872],[1797,6443,1829],{"className":6444},[1877],[1797,6446],{"className":6447,"style":1873},[1872],[1797,6449,6451,6454,6458,6464,6467,6470,6473,6476],{"className":6450},[1849],[1797,6452],{"className":6453,"style":1854},[1853],[1797,6455,6457],{"className":6456},[1858],"300",[1797,6459,6461],{"className":6460},[1858],[1797,6462,2542],{"className":6463},[2782],[1797,6465,5729],{"className":6466},[1858],[1797,6468,1835],{"className":6469},[1893],[1797,6471],{"className":6472,"style":1873},[1872],[1797,6474,2577],{"className":6475},[1877],[1797,6477],{"className":6478,"style":1873},[1872],[1797,6480,6482,6485,6488,6491,6494],{"className":6481},[1849],[1797,6483],{"className":6484,"style":4779},[1853],[1797,6486,5679],{"className":6487},[1858],[1797,6489],{"className":6490,"style":2177},[1872],[1797,6492,6409],{"className":6493},[2181],[1797,6495],{"className":6496,"style":2177},[1872],[1797,6498,6500,6503,6506,6509,6512],{"className":6499},[1849],[1797,6501],{"className":6502,"style":2420},[1853],[1797,6504,6412],{"className":6505},[1858],[1797,6507],{"className":6508,"style":1873},[1872],[1797,6510,2577],{"className":6511},[1877],[1797,6513],{"className":6514,"style":1873},[1872],[1797,6516,6518,6521],{"className":6517},[1849],[1797,6519],{"className":6520,"style":2420},[1853],[1797,6522,6417],{"className":6523},[1858],[1793,6525,6526],{},"The result is conditional on the threshold, exceedance rate, fitted shape, common price level, and independence assumptions.",[1926,6528,6530],{"id":6529},"_7-extremes-are-often-dependent","7. Extremes are often dependent",[1793,6532,6533],{},"One storm can generate thousands of claims. Claim-level independence then understates occurrence aggregation. Decide whether the modelling unit is:",[2946,6535,6536,6539,6542,6545],{},[2949,6537,6538],{},"claim loss;",[2949,6540,6541],{},"policy loss;",[2949,6543,6544],{},"event\u002Foccurrence loss;",[2949,6546,6547],{},"annual aggregate loss.",[1793,6549,6550],{},"Declustering or event identifiers may be required before applying an extreme-value model.",[1926,6552,6554],{"id":6553},"_8-current-case-secondary-perils","8. Current case: secondary perils",[1793,6556,6557],{},"Swiss Re Institute reported 2025 global natural-catastrophe economic losses of about USD 220bn, 49% insured, and attributed 92% of insured losses to secondary perils such as severe convective storms, floods, and wildfires. This is useful for asking:",[2946,6559,6560,6563,6566,6569],{},[2949,6561,6562],{},"should frequency and severity be conditioned on peril and region?",[2949,6564,6565],{},"do events share climate and inflation drivers?",[2949,6567,6568],{},"are annual totals dominated by one event or many medium events?",[2949,6570,6571],{},"what is the effect of policy penetration and reporting coverage?",[1793,6573,6574,6575,6579],{},"The figures do ",[6576,6577,6578],"strong",{},"not"," prove that a Pareto or GPD fits a particular insurer's claims. Distribution choice requires insurer-level, definition-consistent data.",[1926,6581,6583],{"id":6582},"practice","Practice",[5620,6585,6586,6706,6762],{},[2949,6587,6588,6589,5804,6641,6705],{},"For Lomax ",[1797,6590,6592,6611],{"className":6591},[1800],[1797,6593,6595],{"className":6594},[1804],[1806,6596,6597],{"xmlns":1808},[1810,6598,6599,6608],{},[1813,6600,6601,6603,6605],{},[1816,6602,2155],{},[1820,6604,2577],{},[2132,6606,6607],{},"3",[1837,6609,6610],{"encoding":1839},"\\alpha=3",[1797,6612,6614,6632],{"className":6613,"ariaHidden":1845},[1844],[1797,6615,6617,6620,6623,6626,6629],{"className":6616},[1849],[1797,6618],{"className":6619,"style":1920},[1853],[1797,6621,2155],{"className":6622,"style":2238},[1858,1859],[1797,6624],{"className":6625,"style":1873},[1872],[1797,6627,2577],{"className":6628},[1877],[1797,6630],{"className":6631,"style":1873},[1872],[1797,6633,6635,6638],{"className":6634},[1849],[1797,6636],{"className":6637,"style":2420},[1853],[1797,6639,6607],{"className":6640},[1858],[1797,6642,6644,6665],{"className":6643},[1800],[1797,6645,6647],{"className":6646},[1804],[1806,6648,6649],{"xmlns":1808},[1810,6650,6651,6662],{},[1813,6652,6653,6655,6657,6659],{},[1816,6654,2144],{},[1820,6656,2577],{},[1816,6658,5669],{"mathvariant":1993},[2132,6660,6661],{},"20,000",[1837,6663,6664],{"encoding":1839},"\\theta=£20{,}000",[1797,6666,6668,6686],{"className":6667,"ariaHidden":1845},[1844],[1797,6669,6671,6674,6677,6680,6683],{"className":6670},[1849],[1797,6672],{"className":6673,"style":2981},[1853],[1797,6675,2144],{"className":6676,"style":2201},[1858,1859],[1797,6678],{"className":6679,"style":1873},[1872],[1797,6681,2577],{"className":6682},[1877],[1797,6684],{"className":6685,"style":1873},[1872],[1797,6687,6689,6692,6696,6702],{"className":6688},[1849],[1797,6690],{"className":6691,"style":2363},[1853],[1797,6693,6695],{"className":6694},[1858],"£20",[1797,6697,6699],{"className":6698},[1858],[1797,6700,2542],{"className":6701},[2782],[1797,6703,5729],{"className":6704},[1858],", find the mean.",[2949,6707,6708,6709,6761],{},"For GPD ",[1797,6710,6712,6731],{"className":6711},[1800],[1797,6713,6715],{"className":6714},[1804],[1806,6716,6717],{"xmlns":1808},[1810,6718,6719,6728],{},[1813,6720,6721,6723,6725],{},[1816,6722,2350],{},[1820,6724,2577],{},[2132,6726,6727],{},"0.6",[1837,6729,6730],{"encoding":1839},"\\xi=0.6",[1797,6732,6734,6752],{"className":6733,"ariaHidden":1845},[1844],[1797,6735,6737,6740,6743,6746,6749],{"className":6736},[1849],[1797,6738],{"className":6739,"style":2363},[1853],[1797,6741,2350],{"className":6742,"style":2367},[1858,1859],[1797,6744],{"className":6745,"style":1873},[1872],[1797,6747,2577],{"className":6748},[1877],[1797,6750],{"className":6751,"style":1873},[1872],[1797,6753,6755,6758],{"className":6754},[1849],[1797,6756],{"className":6757,"style":2420},[1853],[1797,6759,6727],{"className":6760},[1858],", which of mean and variance exist?",[2949,6763,6764],{},"Why can increasing a GPD threshold make the estimated tail shape less stable even if the model approximation improves?",[6766,6767,6769],"legacy-details",{"title":6768},"Answers",[5620,6770,6771,6887,6996],{},[2949,6772,6773,2647],{},[1797,6774,6776,6809],{"className":6775},[1800],[1797,6777,6779],{"className":6778},[1804],[1806,6780,6781],{"xmlns":1808},[1810,6782,6783,6806],{},[1813,6784,6785,6787,6789,6791,6793,6795,6797,6799,6801,6803],{},[2132,6786,6661],{},[1816,6788,1994],{"mathvariant":1993},[1820,6790,1823],{"stretchy":1822},[2132,6792,6607],{},[1820,6794,1986],{},[2132,6796,2134],{},[1820,6798,1835],{"stretchy":1822},[1820,6800,2577],{},[1816,6802,5669],{"mathvariant":1993},[2132,6804,6805],{},"10,000",[1837,6807,6808],{"encoding":1839},"20{,}000\u002F(3-1)=£10{,}000",[1797,6810,6812,6847,6868],{"className":6811,"ariaHidden":1845},[1844],[1797,6813,6815,6818,6822,6828,6832,6835,6838,6841,6844],{"className":6814},[1849],[1797,6816],{"className":6817,"style":1854},[1853],[1797,6819,6821],{"className":6820},[1858],"20",[1797,6823,6825],{"className":6824},[1858],[1797,6826,2542],{"className":6827},[2782],[1797,6829,6831],{"className":6830},[1858],"000\u002F",[1797,6833,1823],{"className":6834},[1864],[1797,6836,6607],{"className":6837},[1858],[1797,6839],{"className":6840,"style":2177},[1872],[1797,6842,1986],{"className":6843},[2181],[1797,6845],{"className":6846,"style":2177},[1872],[1797,6848,6850,6853,6856,6859,6862,6865],{"className":6849},[1849],[1797,6851],{"className":6852,"style":1854},[1853],[1797,6854,2134],{"className":6855},[1858],[1797,6857,1835],{"className":6858},[1893],[1797,6860],{"className":6861,"style":1873},[1872],[1797,6863,2577],{"className":6864},[1877],[1797,6866],{"className":6867,"style":1873},[1872],[1797,6869,6871,6874,6878,6884],{"className":6870},[1849],[1797,6872],{"className":6873,"style":2363},[1853],[1797,6875,6877],{"className":6876},[1858],"£10",[1797,6879,6881],{"className":6880},[1858],[1797,6882,2542],{"className":6883},[2782],[1797,6885,5729],{"className":6886},[1858],[2949,6888,6889,6890,6942,6943,2647],{},"The mean exists because ",[1797,6891,6893,6911],{"className":6892},[1800],[1797,6894,6896],{"className":6895},[1804],[1806,6897,6898],{"xmlns":1808},[1810,6899,6900,6908],{},[1813,6901,6902,6904,6906],{},[2132,6903,6727],{},[1820,6905,2288],{},[2132,6907,2134],{},[1837,6909,6910],{"encoding":1839},"0.6\u003C1",[1797,6912,6914,6933],{"className":6913,"ariaHidden":1845},[1844],[1797,6915,6917,6921,6924,6927,6930],{"className":6916},[1849],[1797,6918],{"className":6919,"style":6920},[1853],"height:0.6835em;vertical-align:-0.0391em;",[1797,6922,6727],{"className":6923},[1858],[1797,6925],{"className":6926,"style":1873},[1872],[1797,6928,2288],{"className":6929},[1877],[1797,6931],{"className":6932,"style":1873},[1872],[1797,6934,6936,6939],{"className":6935},[1849],[1797,6937],{"className":6938,"style":2420},[1853],[1797,6940,2134],{"className":6941},[1858],"; variance does not because ",[1797,6944,6946,6965],{"className":6945},[1800],[1797,6947,6949],{"className":6948},[1804],[1806,6950,6951],{"xmlns":1808},[1810,6952,6953,6962],{},[1813,6954,6955,6957,6959],{},[2132,6956,6727],{},[1820,6958,3183],{},[2132,6960,6961],{},"0.5",[1837,6963,6964],{"encoding":1839},"0.6\\ge0.5",[1797,6966,6968,6987],{"className":6967,"ariaHidden":1845},[1844],[1797,6969,6971,6975,6978,6981,6984],{"className":6970},[1849],[1797,6972],{"className":6973,"style":6974},[1853],"height:0.7804em;vertical-align:-0.136em;",[1797,6976,6727],{"className":6977},[1858],[1797,6979],{"className":6980,"style":1873},[1872],[1797,6982,3183],{"className":6983},[1877],[1797,6985],{"className":6986,"style":1873},[1872],[1797,6988,6990,6993],{"className":6989},[1849],[1797,6991],{"className":6992,"style":2420},[1853],[1797,6994,6961],{"className":6995},[1858],[2949,6997,6998],{},"Fewer observations exceed the threshold, so sampling and parameter uncertainty increase.",[1926,7000,7002],{"id":7001},"sources-and-further-reading","Sources and further reading",[2946,7004,7005,7013,7020,7026],{},[2949,7006,7007,7008,7012],{},"Pickands, J. (1975), “Statistical Inference Using Extreme Order Statistics,” ",[7009,7010,7011],"em",{},"Annals of Statistics",", 3(1), 119–131.",[2949,7014,7015,7016,7019],{},"Balkema, A. A. and de Haan, L. (1974), “Residual Life Time at Great Age,” ",[7009,7017,7018],{},"Annals of Probability",", 2(5), 792–804.",[2949,7021,7022,7023,2647],{},"Embrechts, P., Klüppelberg, C., and Mikosch, T. (1997), ",[7009,7024,7025],{},"Modelling Extremal Events for Insurance and Finance",[2949,7027,7028],{},[7029,7030,7034],"a",{"href":7031,"rel":7032},"https:\u002F\u002Fwww.swissre.com\u002Fpress-release\u002FWildfires-storms-floods-contribute-to-record-92-of-global-insured-losses-in-2025-says-Swiss-Re-Institute\u002F7b39b1a5-b878-4a55-a5ff-bf5aa561a675",[7033],"nofollow","Swiss Re Institute — natural catastrophes in 2025",{"title":10,"searchDepth":7036,"depth":7036,"links":7037},2,[7038,7039,7040,7041,7042,7043,7044,7045,7046,7047],{"id":1928,"depth":7036,"text":1929},{"id":2432,"depth":7036,"text":2433},{"id":3117,"depth":7036,"text":3118},{"id":4318,"depth":7036,"text":4319},{"id":5606,"depth":7036,"text":5607},{"id":5642,"depth":7036,"text":5643},{"id":6529,"depth":7036,"text":6530},{"id":6553,"depth":7036,"text":6554},{"id":6582,"depth":7036,"text":6583},{"id":7001,"depth":7036,"text":7002},"Model large losses with Weibull, Pareto, and threshold exceedances while making tail uncertainty visible.","md",{"sidebar":7051},{"order":7036},true,{"title":868,"description":7048},"G50p8CoDSVVqCCrHUOOCvRI16wZvmpa9JFr0Ic2x0Yw",[7056,7058],{"title":864,"path":865,"stem":866,"description":7057,"children":-1},"Compare Exponential, Gamma, and lognormal claim models through moments, quantiles, and layer costs.",{"title":872,"path":873,"stem":874,"description":7059,"children":-1},"Model claim frequency with exposure, overdispersion, heterogeneity, and excess zeros.",1785754734772]