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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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20-1h17z",[1814,3157,3158,3161],{"style":2543},[1814,3159],{"className":3160,"style":2525},[2072],[1814,3162,3164,3208,3211,3251],{"className":3163},[1884],[1814,3165,3167],{"className":3166},[2144,2508],[1814,3168,3170,3200],{"className":3169},[2055,2056],[1814,3171,3173,3197],{"className":3172},[2060],[1814,3174,3176,3187],{"className":3175,"style":2781},[2064],[1814,3177,3178,3181],{"style":2521},[1814,3179],{"className":3180,"style":2525},[2072],[1814,3182,3184],{"className":3183},[2077,2078,2079,2080],[1814,3185,1967],{"className":3186,"style":2021},[1884,1885,2080],[1814,3188,3189,3192],{"style":2543},[1814,3190],{"className":3191,"style":2525},[2072],[1814,3193,3194],{},[1814,3195,2115],{"className":3196},[2144,2148,2552],[1814,3198,2088],{"className":3199},[2087],[1814,3201,3203],{"className":3202},[2060],[1814,3204,3206],{"className":3205,"style":2577},[2064],[1814,3207],{},[1814,3209],{"className":3210,"style":1920},[1890],[1814,3212,3214,3217],{"className":3213},[1884],[1814,3215,1978],{"className":3216,"style":2047},[1884,1885],[1814,3218,3220],{"className":3219},[2051],[1814,3221,3223,3243],{"className":3222},[2055,2056],[1814,3224,3226,3240],{"className":3225},[2060],[1814,3227,3229],{"className":3228,"style":2065},[2064],[1814,3230,3231,3234],{"style":2068},[1814,3232],{"className":3233,"style":2073},[2072],[1814,3235,3237],{"className":3236},[2077,2078,2079,2080],[1814,3238,1967],{"className":3239,"style":2021},[1884,1885,2080],[1814,3241,2088],{"className":3242},[2087],[1814,3244,3246],{"className":3245},[2060],[1814,3247,3249],{"className":3248,"style":2095},[2064],[1814,3250],{},[1814,3252,3254,3257],{"className":3253},[1884],[1814,3255,2953],{"className":3256,"style":2047},[1884,1885],[1814,3258,3260],{"className":3259},[2051],[1814,3261,3263,3297],{"className":3262},[2055,2056],[1814,3264,3266,3294],{"className":3265},[2060],[1814,3267,3270,3282],{"className":3268,"style":3269},[2064],"height:0.8641em;",[1814,3271,3273,3276],{"style":3272},"top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;",[1814,3274],{"className":3275,"style":2073},[2072],[1814,3277,3279],{"className":3278},[2077,2078,2079,2080],[1814,3280,1967],{"className":3281,"style":2021},[1884,1885,2080],[1814,3283,3285,3288],{"style":3284},"top:-3.113em;margin-right:0.05em;",[1814,3286],{"className":3287,"style":2073},[2072],[1814,3289,3291],{"className":3290},[2077,2078,2079,2080],[1814,3292,2958],{"className":3293},[1884,2080],[1814,3295,2088],{"className":3296},[2087],[1814,3298,3300],{"className":3299},[2060],[1814,3301,3304],{"className":3302,"style":3303},[2064],"height:0.247em;",[1814,3305],{},[1814,3307,2088],{"className":3308},[2087],[1814,3310,3312],{"className":3311},[2060],[1814,3313,3316],{"className":3314,"style":3315},[2064],"height:1.9501em;",[1814,3317],{},[1814,3319,2088],{"className":3320},[2087],[1814,3322,3324],{"className":3323},[2060],[1814,3325,3328],{"className":3326,"style":3327},[2064],"height:2.6362em;",[1814,3329],{},[1814,3331],{"className":3332,"style":3333},[1890],"margin-right:0.2222em;",[1814,3335,2968],{"className":3336},[3337],"mbin",[1814,3339],{"className":3340,"style":3333},[1890],[1814,3342,3344,3347,3624,3627],{"className":3343},[1875],[1814,3345],{"className":3346,"style":3057},[1879],[1814,3348,3350],{"className":3349},[1924,2707],[1814,3351,3353,3616],{"className":3352},[2055,2056],[1814,3354,3356,3613],{"className":3355},[2060],[1814,3357,3359,3376],{"className":3358,"style":2781},[2064],[1814,3360,3361,3364],{"style":3072},[1814,3362],{"className":3363,"style":2525},[2072],[1814,3365,3367],{"className":3366},[2077,2078,2079,2080],[1814,3368,3370],{"className":3369},[1884,2080],[1814,3371,3373],{"className":3372},[1884,3085,2080],[1814,3374,3011],{"className":3375},[1884,2080],[1814,3377,3378,3381],{"style":2543},[1814,3379],{"className":3380,"style":2525},[2072],[1814,3382,3384],{"className":3383},[1924,2707],[1814,3385,3387,3605],{"className":3386},[2055,2056],[1814,3388,3390,3602],{"className":3389},[2060],[1814,3391,3393,3423],{"className":3392,"style":2781},[2064],[1814,3394,3396,3399],{"className":3395,"style":3110},[3109],[1814,3397],{"className":3398,"style":2525},[2072],[1814,3400,3402,3409,3416],{"className":3401,"style":3118},[3117],[1814,3403,3405],{"className":3404,"style":3123},[3122],[3125,3406,3407],{"xmlns":3127,"width":3128,"height":3129,"viewBox":3130,"preserveAspectRatio":3131},[3133,3408],{"d":3135},[1814,3410,3412],{"className":3411,"style":3123},[3139],[3125,3413,3414],{"xmlns":3127,"width":3128,"height":3129,"viewBox":3130,"preserveAspectRatio":3142},[3133,3415],{"d":3145},[1814,3417,3419],{"className":3418,"style":3123},[3149],[3125,3420,3421],{"xmlns":3127,"width":3128,"height":3129,"viewBox":3130,"preserveAspectRatio":3152},[3133,3422],{"d":3155},[1814,3424,3425,3428],{"style":2543},[1814,3426],{"className":3427,"style":2525},[2072],[1814,3429,3431,3475,3478,3518,3521,3561,3564,3567,3570,3573],{"className":3430},[1884],[1814,3432,3434],{"className":3433},[2144,2508],[1814,3435,3437,3467],{"className":3436},[2055,2056],[1814,3438,3440,3464],{"className":3439},[2060],[1814,3441,3443,3454],{"className":3442,"style":2781},[2064],[1814,3444,3445,3448],{"style":2521},[1814,3446],{"className":3447,"style":2525},[2072],[1814,3449,3451],{"className":3450},[2077,2078,2079,2080],[1814,3452,1967],{"className":3453,"style":2021},[1884,1885,2080],[1814,3455,3456,3459],{"style":2543},[1814,3457],{"className":3458,"style":2525},[2072],[1814,3460,3461],{},[1814,3462,2115],{"className":3463},[2144,2148,2552],[1814,3465,2088],{"className":3466},[2087],[1814,3468,3470],{"className":3469},[2060],[1814,3471,3473],{"className":3472,"style":2577},[2064],[1814,3474],{},[1814,3476],{"className":3477,"style":1920},[1890],[1814,3479,3481,3484],{"className":3480},[1884],[1814,3482,1978],{"className":3483,"style":2047},[1884,1885],[1814,3485,3487],{"className":3486},[2051],[1814,3488,3490,3510],{"className":3489},[2055,2056],[1814,3491,3493,3507],{"className":3492},[2060],[1814,3494,3496],{"className":3495,"style":2065},[2064],[1814,3497,3498,3501],{"style":2068},[1814,3499],{"className":3500,"style":2073},[2072],[1814,3502,3504],{"className":3503},[2077,2078,2079,2080],[1814,3505,1967],{"className":3506,"style":2021},[1884,1885,2080],[1814,3508,2088],{"className":3509},[2087],[1814,3511,3513],{"className":3512},[2060],[1814,3514,3516],{"className":3515,"style":2095},[2064],[1814,3517],{},[1814,3519,1959],{"className":3520},[1909],[1814,3522,3524,3527],{"className":3523},[1884],[1814,3525,2723],{"className":3526},[1884,1885],[1814,3528,3530],{"className":3529},[2051],[1814,3531,3533,3553],{"className":3532},[2055,2056],[1814,3534,3536,3550],{"className":3535},[2060],[1814,3537,3539],{"className":3538,"style":2065},[2064],[1814,3540,3541,3544],{"style":2877},[1814,3542],{"className":3543,"style":2073},[2072],[1814,3545,3547],{"className":3546},[2077,2078,2079,2080],[1814,3548,1967],{"className":3549,"style":2021},[1884,1885,2080],[1814,3551,2088],{"className":3552},[2087],[1814,3554,3556],{"className":3555},[2060],[1814,3557,3559],{"className":3558,"style":2095},[2064],[1814,3560],{},[1814,3562],{"className":3563,"style":3333},[1890],[1814,3565,2997],{"className":3566},[3337],[1814,3568],{"className":3569,"style":3333},[1890],[1814,3571,2723],{"className":3572},[1884,1885],[1814,3574,3576,3579],{"className":3575},[1940],[1814,3577,1970],{"className":3578},[1940],[1814,3580,3582],{"className":3581},[2051],[1814,3583,3585],{"className":3584},[2055],[1814,3586,3588],{"className":3587},[2060],[1814,3589,3591],{"className":3590,"style":3269},[2064],[1814,3592,3593,3596],{"style":3284},[1814,3594],{"className":3595,"style":2073},[2072],[1814,3597,3599],{"className":3598},[2077,2078,2079,2080],[1814,3600,2958],{"className":3601},[1884,2080],[1814,3603,2088],{"className":3604},[2087],[1814,3606,3608],{"className":3607},[2060],[1814,3609,3611],{"className":3610,"style":3315},[2064],[1814,3612],{},[1814,3614,2088],{"className":3615},[2087],[1814,3617,3619],{"className":3618},[2060],[1814,3620,3622],{"className":3621,"style":3327},[2064],[1814,3623],{},[1814,3625],{"className":3626,"style":1920},[1890],[1814,3628,2425],{"className":3629},[1884],[3631,3632,3634],"h3",{"id":3633},"example-ordinary-and-complex-claims","Example: ordinary and complex claims",[1793,3636,3637],{},"Suppose 90% of claims have mean £2,000 and SD £1,000; 10% have mean £20,000 and SD £10,000.",[1814,3639,3641],{"className":3640},[2359],[1814,3642,3644,3696],{"className":3643},[1817],[1814,3645,3647],{"className":3646},[1821],[1823,3648,3649],{"xmlns":1825,"display":2368},[1827,3650,3651,3693],{},[1830,3652,3653,3655,3657,3659,3661,3663,3666,3668,3671,3673,3675,3678,3680,3683,3685,3687,3690],{},[1833,3654,2695],{},[1837,3656,2698],{"stretchy":1842},[1833,3658,2379],{},[1837,3660,2703],{"stretchy":1842},[1837,3662,1964],{},[1845,3664,3665],{},"0.9",[1837,3667,1959],{"stretchy":1842},[1845,3669,3670],{},"2,000",[1837,3672,1970],{"stretchy":1842},[1837,3674,2968],{},[1845,3676,3677],{},"0.1",[1837,3679,1959],{"stretchy":1842},[1845,3681,3682],{},"20,000",[1837,3684,1970],{"stretchy":1842},[1837,3686,1964],{},[1833,3688,3689],{"mathvariant":2424},"£",[1845,3691,3692],{},"3,800.",[1864,3694,3695],{"encoding":1866},"E[X]=0.9(2{,}000)+0.1(20{,}000)=£3{,}800.",[1814,3697,3699,3726,3763,3800],{"className":3698,"ariaHidden":1850},[1871],[1814,3700,3702,3705,3708,3711,3714,3717,3720,3723],{"className":3701},[1875],[1814,3703],{"className":3704,"style":1905},[1879],[1814,3706,2695],{"className":3707,"style":2743},[1884,1885],[1814,3709,2698],{"className":3710},[1909],[1814,3712,2379],{"className":3713,"style":2468},[1884,1885],[1814,3715,2703],{"className":3716},[1940],[1814,3718],{"className":3719,"style":1891},[1890],[1814,3721,1964],{"className":3722},[1895],[1814,3724],{"className":3725,"style":1891},[1890],[1814,3727,3729,3732,3735,3738,3741,3747,3751,3754,3757,3760],{"className":3728},[1875],[1814,3730],{"className":3731,"style":1905},[1879],[1814,3733,3665],{"className":3734},[1884],[1814,3736,1959],{"className":3737},[1909],[1814,3739,2958],{"className":3740},[1884],[1814,3742,3744],{"className":3743},[1884],[1814,3745,1851],{"className":3746},[1916],[1814,3748,3750],{"className":3749},[1884],"000",[1814,3752,1970],{"className":3753},[1940],[1814,3755],{"className":3756,"style":3333},[1890],[1814,3758,2968],{"className":3759},[3337],[1814,3761],{"className":3762,"style":3333},[1890],[1814,3764,3766,3769,3772,3775,3779,3785,3788,3791,3794,3797],{"className":3765},[1875],[1814,3767],{"className":3768,"style":1905},[1879],[1814,3770,3677],{"className":3771},[1884],[1814,3773,1959],{"className":3774},[1909],[1814,3776,3778],{"className":3777},[1884],"20",[1814,3780,3782],{"className":3781},[1884],[1814,3783,1851],{"className":3784},[1916],[1814,3786,3750],{"className":3787},[1884],[1814,3789,1970],{"className":3790},[1940],[1814,3792],{"className":3793,"style":1891},[1890],[1814,3795,1964],{"className":3796},[1895],[1814,3798],{"className":3799,"style":1891},[1890],[1814,3801,3803,3806,3810,3816],{"className":3802},[1875],[1814,3804],{"className":3805,"style":2312},[1879],[1814,3807,3809],{"className":3808},[1884],"£3",[1814,3811,3813],{"className":3812},[1884],[1814,3814,1851],{"className":3815},[1916],[1814,3817,3819],{"className":3818},[1884],"800.",[1793,3821,3822],{},"The mixture SD is about £6,329. A single “average claim” hides both the rare high-cost class and the uncertainty over class membership.",[1806,3824,3826],{"id":3825},"_2-observed-segmentation-versus-latent-class","2. Observed segmentation versus latent class",[3828,3829,3830,3849],"table",{},[3831,3832,3833],"thead",{},[3834,3835,3836,3840,3843,3846],"tr",{},[3837,3838,3839],"th",{},"Approach",[3837,3841,3842],{},"Example",[3837,3844,3845],{},"Advantage",[3837,3847,3848],{},"Risk",[3850,3851,3852,3867,3881],"tbody",{},[3834,3853,3854,3858,3861,3864],{},[3855,3856,3857],"td",{},"observed segmentation",[3855,3859,3860],{},"vehicle type or injury indicator",[3855,3862,3863],{},"interpretable and actionable",[3855,3865,3866],{},"coding changes; small cells",[3834,3868,3869,3872,3875,3878],{},[3855,3870,3871],{},"latent finite mixture",[3855,3873,3874],{},"unobserved simple\u002Fcomplex claim",[3855,3876,3877],{},"flexible distribution shape",[3855,3879,3880],{},"classes may not have unique meaning",[3834,3882,3883,3886,3889,3892],{},[3855,3884,3885],{},"hierarchical\u002Frandom effect",[3855,3887,3888],{},"region or policyholder risk varies continuously",[3855,3890,3891],{},"partial pooling",[3855,3893,3894],{},"distribution of random effects matters",[1793,3896,3897],{},"Prefer observed, causally relevant information when available. Latent classes are mathematical devices unless external evidence gives them operational meaning.",[1806,3899,3901],{"id":3900},"_3-continuous-mixture-poissongamma-gives-negative-binomial","3. Continuous mixture: Poisson–Gamma gives negative binomial",[1793,3903,3904],{},"Suppose conditional claim count is Poisson:",[1814,3906,3908],{"className":3907},[2359],[1814,3909,3911,3948],{"className":3910},[1817],[1814,3912,3914],{"className":3913},[1821],[1823,3915,3916],{"xmlns":1825,"display":2368},[1827,3917,3918,3945],{},[1830,3919,3920,3923,3926,3929,3932,3935,3937,3939,3941,3943],{},[1833,3921,3922],{},"N",[1837,3924,3925],{},"∣",[1833,3927,3928],{"mathvariant":2424},"Λ",[1837,3930,3931],{},"∼",[1833,3933,3934],{"mathvariant":2424},"Poisson",[1837,3936,2921],{},[1837,3938,1959],{"stretchy":1842},[1833,3940,3928],{"mathvariant":2424},[1837,3942,1970],{"stretchy":1842},[1837,3944,1851],{"separator":1850},[1864,3946,3947],{"encoding":1866},"N\\mid\\Lambda\\sim\\operatorname{Poisson}(\\Lambda),",[1814,3949,3951,3970,3989],{"className":3950,"ariaHidden":1850},[1871],[1814,3952,3954,3957,3961,3964,3967],{"className":3953},[1875],[1814,3955],{"className":3956,"style":1905},[1879],[1814,3958,3922],{"className":3959,"style":3960},[1884,1885],"margin-right:0.109em;",[1814,3962],{"className":3963,"style":1891},[1890],[1814,3965,3925],{"className":3966},[1895],[1814,3968],{"className":3969,"style":1891},[1890],[1814,3971,3973,3977,3980,3983,3986],{"className":3972},[1875],[1814,3974],{"className":3975,"style":3976},[1879],"height:0.6833em;",[1814,3978,3928],{"className":3979},[1884],[1814,3981],{"className":3982,"style":1891},[1890],[1814,3984,3931],{"className":3985},[1895],[1814,3987],{"className":3988,"style":1891},[1890],[1814,3990,3992,3995,4001,4004,4007,4010],{"className":3991},[1875],[1814,3993],{"className":3994,"style":1905},[1879],[1814,3996,3998],{"className":3997},[2144],[1814,3999,3934],{"className":4000},[1884,3032],[1814,4002,1959],{"className":4003},[1909],[1814,4005,3928],{"className":4006},[1884],[1814,4008,1970],{"className":4009},[1940],[1814,4011,1851],{"className":4012},[1916],[1793,4014,4015],{},"but latent rates differ across risks:",[1814,4017,4019],{"className":4018},[2359],[1814,4020,4022,4076],{"className":4021},[1817],[1814,4023,4025],{"className":4024},[1821],[1823,4026,4027],{"xmlns":1825,"display":2368},[1827,4028,4029,4073],{},[1830,4030,4031,4033,4035,4037,4039,4041,4043,4045,4048,4050,4052,4054,4056,4058,4060,4071],{},[1833,4032,2695],{},[1837,4034,2698],{"stretchy":1842},[1833,4036,3928],{"mathvariant":2424},[1837,4038,2703],{"stretchy":1842},[1837,4040,1964],{},[1833,4042,2723],{},[1837,4044,1851],{"separator":1850},[1890,4046],{"width":4047},"2em",[1833,4049,2918],{"mathvariant":2424},[1837,4051,2921],{},[1837,4053,1959],{"stretchy":1842},[1833,4055,3928],{"mathvariant":2424},[1837,4057,1970],{"stretchy":1842},[1837,4059,1964],{},[4061,4062,4063,4069],"mfrac",{},[3001,4064,4065,4067],{},[1833,4066,2723],{},[1845,4068,2958],{},[1833,4070,1967],{},[1833,4072,2425],{"mathvariant":2424},[1864,4074,4075],{"encoding":1866},"E[\\Lambda]=\\mu,\n\\qquad\n\\operatorname{Var}(\\Lambda)=\\frac{\\mu^2}{k}.",[1814,4077,4079,4106,4149],{"className":4078,"ariaHidden":1850},[1871],[1814,4080,4082,4085,4088,4091,4094,4097,4100,4103],{"className":4081},[1875],[1814,4083],{"className":4084,"style":1905},[1879],[1814,4086,2695],{"className":4087,"style":2743},[1884,1885],[1814,4089,2698],{"className":4090},[1909],[1814,4092,3928],{"className":4093},[1884],[1814,4095,2703],{"className":4096},[1940],[1814,4098],{"className":4099,"style":1891},[1890],[1814,4101,1964],{"className":4102},[1895],[1814,4104],{"className":4105,"style":1891},[1890],[1814,4107,4109,4112,4115,4118,4122,4125,4131,4134,4137,4140,4143,4146],{"className":4108},[1875],[1814,4110],{"className":4111,"style":1905},[1879],[1814,4113,2723],{"className":4114},[1884,1885],[1814,4116,1851],{"className":4117},[1916],[1814,4119],{"className":4120,"style":4121},[1890],"margin-right:2em;",[1814,4123],{"className":4124,"style":1920},[1890],[1814,4126,4128],{"className":4127},[2144],[1814,4129,2918],{"className":4130},[1884,3032],[1814,4132,1959],{"className":4133},[1909],[1814,4135,3928],{"className":4136},[1884],[1814,4138,1970],{"className":4139},[1940],[1814,4141],{"className":4142,"style":1891},[1890],[1814,4144,1964],{"className":4145},[1895],[1814,4147],{"className":4148,"style":1891},[1890],[1814,4150,4152,4156,4255],{"className":4151},[1875],[1814,4153],{"className":4154,"style":4155},[1879],"height:2.1771em;vertical-align:-0.686em;",[1814,4157,4159,4163,4252],{"className":4158},[1884],[1814,4160],{"className":4161},[1909,4162],"nulldelimiter",[1814,4164,4166],{"className":4165},[4061],[1814,4167,4169,4243],{"className":4168},[2055,2056],[1814,4170,4172,4240],{"className":4171},[2060],[1814,4173,4176,4189,4200],{"className":4174,"style":4175},[2064],"height:1.4911em;",[1814,4177,4179,4183],{"style":4178},"top:-2.314em;",[1814,4180],{"className":4181,"style":4182},[2072],"height:3em;",[1814,4184,4186],{"className":4185},[1884],[1814,4187,1967],{"className":4188,"style":2021},[1884,1885],[1814,4190,4192,4195],{"style":4191},"top:-3.23em;",[1814,4193],{"className":4194,"style":4182},[2072],[1814,4196],{"className":4197,"style":4199},[4198],"frac-line","border-bottom-width:0.04em;",[1814,4201,4203,4206],{"style":4202},"top:-3.677em;",[1814,4204],{"className":4205,"style":4182},[2072],[1814,4207,4209],{"className":4208},[1884],[1814,4210,4212,4215],{"className":4211},[1884],[1814,4213,2723],{"className":4214},[1884,1885],[1814,4216,4218],{"className":4217},[2051],[1814,4219,4221],{"className":4220},[2055],[1814,4222,4224],{"className":4223},[2060],[1814,4225,4228],{"className":4226,"style":4227},[2064],"height:0.8141em;",[1814,4229,4231,4234],{"style":4230},"top:-3.063em;margin-right:0.05em;",[1814,4232],{"className":4233,"style":2073},[2072],[1814,4235,4237],{"className":4236},[2077,2078,2079,2080],[1814,4238,2958],{"className":4239},[1884,2080],[1814,4241,2088],{"className":4242},[2087],[1814,4244,4246],{"className":4245},[2060],[1814,4247,4250],{"className":4248,"style":4249},[2064],"height:0.686em;",[1814,4251],{},[1814,4253],{"className":4254},[1940,4162],[1814,4256,2425],{"className":4257},[1884],[1793,4259,4260],{},"Gamma mixing produces a negative-binomial marginal count with",[1814,4262,4264],{"className":4263},[2359],[1814,4265,4267,4323],{"className":4266},[1817],[1814,4268,4270],{"className":4269},[1821],[1823,4271,4272],{"xmlns":1825,"display":2368},[1827,4273,4274,4320],{},[1830,4275,4276,4278,4280,4282,4284,4286,4288,4290,4292,4294,4296,4298,4300,4302,4304,4306,4308,4318],{},[1833,4277,2695],{},[1837,4279,2698],{"stretchy":1842},[1833,4281,3922],{},[1837,4283,2703],{"stretchy":1842},[1837,4285,1964],{},[1833,4287,2723],{},[1837,4289,1851],{"separator":1850},[1890,4291],{"width":4047},[1833,4293,2918],{"mathvariant":2424},[1837,4295,2921],{},[1837,4297,1959],{"stretchy":1842},[1833,4299,3922],{},[1837,4301,1970],{"stretchy":1842},[1837,4303,1964],{},[1833,4305,2723],{},[1837,4307,2968],{},[4061,4309,4310,4316],{},[3001,4311,4312,4314],{},[1833,4313,2723],{},[1845,4315,2958],{},[1833,4317,1967],{},[1833,4319,2425],{"mathvariant":2424},[1864,4321,4322],{"encoding":1866},"E[N]=\\mu,\n\\qquad\n\\operatorname{Var}(N)=\\mu+\\frac{\\mu^2}{k}.",[1814,4324,4326,4353,4395,4414],{"className":4325,"ariaHidden":1850},[1871],[1814,4327,4329,4332,4335,4338,4341,4344,4347,4350],{"className":4328},[1875],[1814,4330],{"className":4331,"style":1905},[1879],[1814,4333,2695],{"className":4334,"style":2743},[1884,1885],[1814,4336,2698],{"className":4337},[1909],[1814,4339,3922],{"className":4340,"style":3960},[1884,1885],[1814,4342,2703],{"className":4343},[1940],[1814,4345],{"className":4346,"style":1891},[1890],[1814,4348,1964],{"className":4349},[1895],[1814,4351],{"className":4352,"style":1891},[1890],[1814,4354,4356,4359,4362,4365,4368,4371,4377,4380,4383,4386,4389,4392],{"className":4355},[1875],[1814,4357],{"className":4358,"style":1905},[1879],[1814,4360,2723],{"className":4361},[1884,1885],[1814,4363,1851],{"className":4364},[1916],[1814,4366],{"className":4367,"style":4121},[1890],[1814,4369],{"className":4370,"style":1920},[1890],[1814,4372,4374],{"className":4373},[2144],[1814,4375,2918],{"className":4376},[1884,3032],[1814,4378,1959],{"className":4379},[1909],[1814,4381,3922],{"className":4382,"style":3960},[1884,1885],[1814,4384,1970],{"className":4385},[1940],[1814,4387],{"className":4388,"style":1891},[1890],[1814,4390,1964],{"className":4391},[1895],[1814,4393],{"className":4394,"style":1891},[1890],[1814,4396,4398,4402,4405,4408,4411],{"className":4397},[1875],[1814,4399],{"className":4400,"style":4401},[1879],"height:0.7778em;vertical-align:-0.1944em;",[1814,4403,2723],{"className":4404},[1884,1885],[1814,4406],{"className":4407,"style":3333},[1890],[1814,4409,2968],{"className":4410},[3337],[1814,4412],{"className":4413,"style":3333},[1890],[1814,4415,4417,4420,4508],{"className":4416},[1875],[1814,4418],{"className":4419,"style":4155},[1879],[1814,4421,4423,4426,4505],{"className":4422},[1884],[1814,4424],{"className":4425},[1909,4162],[1814,4427,4429],{"className":4428},[4061],[1814,4430,4432,4497],{"className":4431},[2055,2056],[1814,4433,4435,4494],{"className":4434},[2060],[1814,4436,4438,4449,4457],{"className":4437,"style":4175},[2064],[1814,4439,4440,4443],{"style":4178},[1814,4441],{"className":4442,"style":4182},[2072],[1814,4444,4446],{"className":4445},[1884],[1814,4447,1967],{"className":4448,"style":2021},[1884,1885],[1814,4450,4451,4454],{"style":4191},[1814,4452],{"className":4453,"style":4182},[2072],[1814,4455],{"className":4456,"style":4199},[4198],[1814,4458,4459,4462],{"style":4202},[1814,4460],{"className":4461,"style":4182},[2072],[1814,4463,4465],{"className":4464},[1884],[1814,4466,4468,4471],{"className":4467},[1884],[1814,4469,2723],{"className":4470},[1884,1885],[1814,4472,4474],{"className":4473},[2051],[1814,4475,4477],{"className":4476},[2055],[1814,4478,4480],{"className":4479},[2060],[1814,4481,4483],{"className":4482,"style":4227},[2064],[1814,4484,4485,4488],{"style":4230},[1814,4486],{"className":4487,"style":2073},[2072],[1814,4489,4491],{"className":4490},[2077,2078,2079,2080],[1814,4492,2958],{"className":4493},[1884,2080],[1814,4495,2088],{"className":4496},[2087],[1814,4498,4500],{"className":4499},[2060],[1814,4501,4503],{"className":4502,"style":4249},[2064],[1814,4504],{},[1814,4506],{"className":4507},[1940,4162],[1814,4509,2425],{"className":4510},[1884],[1793,4512,4513],{},"The first variance term is Poisson process variation; the second is heterogeneity.",[4515,4516],"web-r",{"code64":4517,"layout":4518,"locale":7,"title":4519},"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","vertical","Poisson–Gamma heterogeneity creates overdispersion",[1806,4521,4523],{"id":4522},"_4-mixture-is-not-compound-aggregation","4. Mixture is not compound aggregation",[1793,4525,4526],{},"Do not confuse:",[1814,4528,4530],{"className":4529},[2359],[1814,4531,4533,4559],{"className":4532},[1817],[1814,4534,4536],{"className":4535},[1821],[1823,4537,4538],{"xmlns":1825,"display":2368},[1827,4539,4540,4556],{},[1830,4541,4542,4545,4547,4549],{},[2963,4543,4544],{},"mixture: ",[1833,4546,2379],{},[1837,4548,3931],{},[1974,4550,4551,4554],{},[1833,4552,4553],{},"F",[1833,4555,1835],{},[1864,4557,4558],{"encoding":1866},"\\text{mixture: } X\\sim F_Z",[1814,4560,4562,4586],{"className":4561,"ariaHidden":1850},[1871],[1814,4563,4565,4568,4574,4577,4580,4583],{"className":4564},[1875],[1814,4566],{"className":4567,"style":3976},[1879],[1814,4569,4571],{"className":4570},[1884,3085],[1814,4572,4544],{"className":4573},[1884],[1814,4575,2379],{"className":4576,"style":2468},[1884,1885],[1814,4578],{"className":4579,"style":1891},[1890],[1814,4581,3931],{"className":4582},[1895],[1814,4584],{"className":4585,"style":1891},[1890],[1814,4587,4589,4593],{"className":4588},[1875],[1814,4590],{"className":4591,"style":4592},[1879],"height:0.8333em;vertical-align:-0.15em;",[1814,4594,4596,4599],{"className":4595},[1884],[1814,4597,4553],{"className":4598,"style":1996},[1884,1885],[1814,4600,4602],{"className":4601},[2051],[1814,4603,4605,4626],{"className":4604},[2055,2056],[1814,4606,4608,4623],{"className":4607},[2060],[1814,4609,4611],{"className":4610,"style":2456},[2064],[1814,4612,4614,4617],{"style":4613},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[1814,4615],{"className":4616,"style":2073},[2072],[1814,4618,4620],{"className":4619},[2077,2078,2079,2080],[1814,4621,1835],{"className":4622,"style":1886},[1884,1885,2080],[1814,4624,2088],{"className":4625},[2087],[1814,4627,4629],{"className":4628},[2060],[1814,4630,4632],{"className":4631,"style":2095},[2064],[1814,4633],{},[1793,4635,4636],{},"with",[1814,4638,4640],{"className":4639},[2359],[1814,4641,4643,4684],{"className":4642},[1817],[1814,4644,4646],{"className":4645},[1821],[1823,4647,4648],{"xmlns":1825,"display":2368},[1827,4649,4650,4681],{},[1830,4651,4652,4655,4658,4660,4666,4668,4671,4673,4679],{},[2963,4653,4654],{},"compound sum: ",[1833,4656,4657],{},"S",[1837,4659,1964],{},[1974,4661,4662,4664],{},[1833,4663,2379],{},[1845,4665,1847],{},[1837,4667,2968],{},[1837,4669,4670],{},"⋯",[1837,4672,2968],{},[1974,4674,4675,4677],{},[1833,4676,2379],{},[1833,4678,3922],{},[1833,4680,2425],{"mathvariant":2424},[1864,4682,4683],{"encoding":1866},"\\text{compound sum: } S=X_1+\\cdots+X_N.",[1814,4685,4687,4711,4768,4787],{"className":4686,"ariaHidden":1850},[1871],[1814,4688,4690,4693,4699,4702,4705,4708],{"className":4689},[1875],[1814,4691],{"className":4692,"style":2312},[1879],[1814,4694,4696],{"className":4695},[1884,3085],[1814,4697,4654],{"className":4698},[1884],[1814,4700,4657],{"className":4701,"style":2743},[1884,1885],[1814,4703],{"className":4704,"style":1891},[1890],[1814,4706,1964],{"className":4707},[1895],[1814,4709],{"className":4710,"style":1891},[1890],[1814,4712,4714,4717,4759,4762,4765],{"className":4713},[1875],[1814,4715],{"className":4716,"style":4592},[1879],[1814,4718,4720,4723],{"className":4719},[1884],[1814,4721,2379],{"className":4722,"style":2468},[1884,1885],[1814,4724,4726],{"className":4725},[2051],[1814,4727,4729,4751],{"className":4728},[2055,2056],[1814,4730,4732,4748],{"className":4731},[2060],[1814,4733,4736],{"className":4734,"style":4735},[2064],"height:0.3011em;",[1814,4737,4739,4742],{"style":4738},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[1814,4740],{"className":4741,"style":2073},[2072],[1814,4743,4745],{"className":4744},[2077,2078,2079,2080],[1814,4746,1847],{"className":4747},[1884,2080],[1814,4749,2088],{"className":4750},[2087],[1814,4752,4754],{"className":4753},[2060],[1814,4755,4757],{"className":4756,"style":2095},[2064],[1814,4758],{},[1814,4760],{"className":4761,"style":3333},[1890],[1814,4763,2968],{"className":4764},[3337],[1814,4766],{"className":4767,"style":3333},[1890],[1814,4769,4771,4775,4778,4781,4784],{"className":4770},[1875],[1814,4772],{"className":4773,"style":4774},[1879],"height:0.6667em;vertical-align:-0.0833em;",[1814,4776,4670],{"className":4777},[1924],[1814,4779],{"className":4780,"style":3333},[1890],[1814,4782,2968],{"className":4783},[3337],[1814,4785],{"className":4786,"style":3333},[1890],[1814,4788,4790,4793,4833],{"className":4789},[1875],[1814,4791],{"className":4792,"style":4592},[1879],[1814,4794,4796,4799],{"className":4795},[1884],[1814,4797,2379],{"className":4798,"style":2468},[1884,1885],[1814,4800,4802],{"className":4801},[2051],[1814,4803,4805,4825],{"className":4804},[2055,2056],[1814,4806,4808,4822],{"className":4807},[2060],[1814,4809,4811],{"className":4810,"style":2456},[2064],[1814,4812,4813,4816],{"style":4738},[1814,4814],{"className":4815,"style":2073},[2072],[1814,4817,4819],{"className":4818},[2077,2078,2079,2080],[1814,4820,3922],{"className":4821,"style":3960},[1884,1885,2080],[1814,4823,2088],{"className":4824},[2087],[1814,4826,4828],{"className":4827},[2060],[1814,4829,4831],{"className":4830,"style":2095},[2064],[1814,4832],{},[1814,4834,2425],{"className":4835},[1884],[1793,4837,4838],{},"A mixture selects a distribution or parameter state; a compound model sums a random number of losses. An insurance portfolio can require both.",[1806,4840,4842],{"id":4841},"_5-changing-mix-can-imitate-inflation","5. Changing mix can imitate inflation",[1793,4844,4845],{},"Suppose group-specific severities stay constant, but the high-cost group's weight rises from 10% to 20%:",[1814,4847,4849],{"className":4848},[2359],[1814,4850,4852,4889],{"className":4851},[1817],[1814,4853,4855],{"className":4854},[1821],[1823,4856,4857],{"xmlns":1825,"display":2368},[1827,4858,4859,4886],{},[1830,4860,4861,4863,4865,4867,4869,4871,4873,4875,4877,4879,4881,4884],{},[1845,4862,3665],{},[1837,4864,1959],{"stretchy":1842},[1845,4866,3670],{},[1837,4868,1970],{"stretchy":1842},[1837,4870,2968],{},[1845,4872,3677],{},[1837,4874,1959],{"stretchy":1842},[1845,4876,3682],{},[1837,4878,1970],{"stretchy":1842},[1837,4880,1964],{},[1845,4882,4883],{},"3,800",[1837,4885,1851],{"separator":1850},[1864,4887,4888],{"encoding":1866},"0.9(2{,}000)+0.1(20{,}000)=3{,}800,",[1814,4890,4892,4928,4964],{"className":4891,"ariaHidden":1850},[1871],[1814,4893,4895,4898,4901,4904,4907,4913,4916,4919,4922,4925],{"className":4894},[1875],[1814,4896],{"className":4897,"style":1905},[1879],[1814,4899,3665],{"className":4900},[1884],[1814,4902,1959],{"className":4903},[1909],[1814,4905,2958],{"className":4906},[1884],[1814,4908,4910],{"className":4909},[1884],[1814,4911,1851],{"className":4912},[1916],[1814,4914,3750],{"className":4915},[1884],[1814,4917,1970],{"className":4918},[1940],[1814,4920],{"className":4921,"style":3333},[1890],[1814,4923,2968],{"className":4924},[3337],[1814,4926],{"className":4927,"style":3333},[1890],[1814,4929,4931,4934,4937,4940,4943,4949,4952,4955,4958,4961],{"className":4930},[1875],[1814,4932],{"className":4933,"style":1905},[1879],[1814,4935,3677],{"className":4936},[1884],[1814,4938,1959],{"className":4939},[1909],[1814,4941,3778],{"className":4942},[1884],[1814,4944,4946],{"className":4945},[1884],[1814,4947,1851],{"className":4948},[1916],[1814,4950,3750],{"className":4951},[1884],[1814,4953,1970],{"className":4954},[1940],[1814,4956],{"className":4957,"style":1891},[1890],[1814,4959,1964],{"className":4960},[1895],[1814,4962],{"className":4963,"style":1891},[1890],[1814,4965,4967,4971,4975,4981,4985],{"className":4966},[1875],[1814,4968],{"className":4969,"style":4970},[1879],"height:0.8389em;vertical-align:-0.1944em;",[1814,4972,4974],{"className":4973},[1884],"3",[1814,4976,4978],{"className":4977},[1884],[1814,4979,1851],{"className":4980},[1916],[1814,4982,4984],{"className":4983},[1884],"800",[1814,4986,1851],{"className":4987},[1916],[1814,4989,4991],{"className":4990},[2359],[1814,4992,4994,5031],{"className":4993},[1817],[1814,4995,4997],{"className":4996},[1821],[1823,4998,4999],{"xmlns":1825,"display":2368},[1827,5000,5001,5028],{},[1830,5002,5003,5006,5008,5010,5012,5014,5017,5019,5021,5023,5025],{},[1845,5004,5005],{},"0.8",[1837,5007,1959],{"stretchy":1842},[1845,5009,3670],{},[1837,5011,1970],{"stretchy":1842},[1837,5013,2968],{},[1845,5015,5016],{},"0.2",[1837,5018,1959],{"stretchy":1842},[1845,5020,3682],{},[1837,5022,1970],{"stretchy":1842},[1837,5024,1964],{},[1845,5026,5027],{},"5,600.",[1864,5029,5030],{"encoding":1866},"0.8(2{,}000)+0.2(20{,}000)=5{,}600.",[1814,5032,5034,5070,5106],{"className":5033,"ariaHidden":1850},[1871],[1814,5035,5037,5040,5043,5046,5049,5055,5058,5061,5064,5067],{"className":5036},[1875],[1814,5038],{"className":5039,"style":1905},[1879],[1814,5041,5005],{"className":5042},[1884],[1814,5044,1959],{"className":5045},[1909],[1814,5047,2958],{"className":5048},[1884],[1814,5050,5052],{"className":5051},[1884],[1814,5053,1851],{"className":5054},[1916],[1814,5056,3750],{"className":5057},[1884],[1814,5059,1970],{"className":5060},[1940],[1814,5062],{"className":5063,"style":3333},[1890],[1814,5065,2968],{"className":5066},[3337],[1814,5068],{"className":5069,"style":3333},[1890],[1814,5071,5073,5076,5079,5082,5085,5091,5094,5097,5100,5103],{"className":5072},[1875],[1814,5074],{"className":5075,"style":1905},[1879],[1814,5077,5016],{"className":5078},[1884],[1814,5080,1959],{"className":5081},[1909],[1814,5083,3778],{"className":5084},[1884],[1814,5086,5088],{"className":5087},[1884],[1814,5089,1851],{"className":5090},[1916],[1814,5092,3750],{"className":5093},[1884],[1814,5095,1970],{"className":5096},[1940],[1814,5098],{"className":5099,"style":1891},[1890],[1814,5101,1964],{"className":5102},[1895],[1814,5104],{"className":5105,"style":1891},[1890],[1814,5107,5109,5112,5116,5122],{"className":5108},[1875],[1814,5110],{"className":5111,"style":4970},[1879],[1814,5113,5115],{"className":5114},[1884],"5",[1814,5117,5119],{"className":5118},[1884],[1814,5120,1851],{"className":5121},[1916],[1814,5123,5125],{"className":5124},[1884],"600.",[1793,5127,5128],{},"Portfolio mean rises 47.4% with no within-group inflation. Severity trends should therefore be decomposed into price, coverage, and mix effects.",[1806,5130,5132],{"id":5131},"_6-identifiability-and-stability","6. Identifiability and stability",[1793,5134,5135],{},"Mixture likelihoods can have multiple local optima, near-empty components, and label switching. A component with tiny weight and huge mean can dominate the tail. Use:",[5137,5138,5139,5143,5146,5149,5152],"ul",{},[5140,5141,5142],"li",{},"several initial values and convergence checks;",[5140,5144,5145],{},"minimum-volume or regularisation rules justified before seeing results;",[5140,5147,5148],{},"holdout performance and tail sensitivity;",[5140,5150,5151],{},"stability across valuation dates;",[5140,5153,5154],{},"comparison with simpler observed segmentation.",[5156,5157,5159],"warning",{"title":5158},"Flexibility is not evidence","A two-component model can fit many shapes. It does not prove that two real claim populations exist, nor that their fitted weights will remain stable.",[1806,5161,5163],{"id":5162},"practice","Practice",[5165,5166,5167,5170,5228],"ol",{},[5140,5168,5169],{},"A 70\u002F30 mixture has component means 1 and 6. Find the mixture mean.",[5140,5171,5172,5173,5227],{},"In a Poisson–Gamma mixture, what happens as ",[1814,5174,5176,5196],{"className":5175},[1817],[1814,5177,5179],{"className":5178},[1821],[1823,5180,5181],{"xmlns":1825},[1827,5182,5183,5193],{},[1830,5184,5185,5187,5190],{},[1833,5186,1967],{},[1837,5188,5189],{},"→",[1833,5191,5192],{"mathvariant":2424},"∞",[1864,5194,5195],{"encoding":1866},"k\\to\\infty",[1814,5197,5199,5217],{"className":5198,"ariaHidden":1850},[1871],[1814,5200,5202,5205,5208,5211,5214],{"className":5201},[1875],[1814,5203],{"className":5204,"style":2276},[1879],[1814,5206,1967],{"className":5207,"style":2021},[1884,1885],[1814,5209],{"className":5210,"style":1891},[1890],[1814,5212,5189],{"className":5213},[1895],[1814,5215],{"className":5216,"style":1891},[1890],[1814,5218,5220,5224],{"className":5219},[1875],[1814,5221],{"className":5222,"style":5223},[1879],"height:0.4306em;",[1814,5225,5192],{"className":5226},[1884],"?",[5140,5229,5230],{},"Why can changing class weights bias a trend estimated from the unsegmented mean?",[5232,5233,5235],"legacy-details",{"title":5234},"Answers",[5165,5236,5237,5346,5515],{},[5140,5238,5239,2425],{},[1814,5240,5242,5280],{"className":5241},[1817],[1814,5243,5245],{"className":5244},[1821],[1823,5246,5247],{"xmlns":1825},[1827,5248,5249,5277],{},[1830,5250,5251,5254,5256,5258,5260,5262,5265,5267,5270,5272,5274],{},[1845,5252,5253],{},"0.7",[1837,5255,1959],{"stretchy":1842},[1845,5257,1847],{},[1837,5259,1970],{"stretchy":1842},[1837,5261,2968],{},[1845,5263,5264],{},"0.3",[1837,5266,1959],{"stretchy":1842},[1845,5268,5269],{},"6",[1837,5271,1970],{"stretchy":1842},[1837,5273,1964],{},[1845,5275,5276],{},"2.5",[1864,5278,5279],{"encoding":1866},"0.7(1)+0.3(6)=2.5",[1814,5281,5283,5310,5337],{"className":5282,"ariaHidden":1850},[1871],[1814,5284,5286,5289,5292,5295,5298,5301,5304,5307],{"className":5285},[1875],[1814,5287],{"className":5288,"style":1905},[1879],[1814,5290,5253],{"className":5291},[1884],[1814,5293,1959],{"className":5294},[1909],[1814,5296,1847],{"className":5297},[1884],[1814,5299,1970],{"className":5300},[1940],[1814,5302],{"className":5303,"style":3333},[1890],[1814,5305,2968],{"className":5306},[3337],[1814,5308],{"className":5309,"style":3333},[1890],[1814,5311,5313,5316,5319,5322,5325,5328,5331,5334],{"className":5312},[1875],[1814,5314],{"className":5315,"style":1905},[1879],[1814,5317,5264],{"className":5318},[1884],[1814,5320,1959],{"className":5321},[1909],[1814,5323,5269],{"className":5324},[1884],[1814,5326,1970],{"className":5327},[1940],[1814,5329],{"className":5330,"style":1891},[1890],[1814,5332,1964],{"className":5333},[1895],[1814,5335],{"className":5336,"style":1891},[1890],[1814,5338,5340,5343],{"className":5339},[1875],[1814,5341],{"className":5342,"style":2246},[1879],[1814,5344,5276],{"className":5345},[1884],[5140,5347,5348,5484,5485,2425],{},[1814,5349,5351,5391],{"className":5350},[1817],[1814,5352,5354],{"className":5353},[1821],[1823,5355,5356],{"xmlns":1825},[1827,5357,5358,5388],{},[1830,5359,5360,5362,5364,5366,5368,5370,5372,5378,5381,5383,5385],{},[1833,5361,2918],{"mathvariant":2424},[1837,5363,2921],{},[1837,5365,1959],{"stretchy":1842},[1833,5367,3928],{"mathvariant":2424},[1837,5369,1970],{"stretchy":1842},[1837,5371,1964],{},[3001,5373,5374,5376],{},[1833,5375,2723],{},[1845,5377,2958],{},[1833,5379,5380],{"mathvariant":2424},"\u002F",[1833,5382,1967],{},[1837,5384,5189],{},[1845,5386,5387],{},"0",[1864,5389,5390],{"encoding":1866},"\\operatorname{Var}(\\Lambda)=\\mu^2\u002Fk\\to0",[1814,5392,5394,5424,5475],{"className":5393,"ariaHidden":1850},[1871],[1814,5395,5397,5400,5406,5409,5412,5415,5418,5421],{"className":5396},[1875],[1814,5398],{"className":5399,"style":1905},[1879],[1814,5401,5403],{"className":5402},[2144],[1814,5404,2918],{"className":5405},[1884,3032],[1814,5407,1959],{"className":5408},[1909],[1814,5410,3928],{"className":5411},[1884],[1814,5413,1970],{"className":5414},[1940],[1814,5416],{"className":5417,"style":1891},[1890],[1814,5419,1964],{"className":5420},[1895],[1814,5422],{"className":5423,"style":1891},[1890],[1814,5425,5427,5431,5460,5463,5466,5469,5472],{"className":5426},[1875],[1814,5428],{"className":5429,"style":5430},[1879],"height:1.0641em;vertical-align:-0.25em;",[1814,5432,5434,5437],{"className":5433},[1884],[1814,5435,2723],{"className":5436},[1884,1885],[1814,5438,5440],{"className":5439},[2051],[1814,5441,5443],{"className":5442},[2055],[1814,5444,5446],{"className":5445},[2060],[1814,5447,5449],{"className":5448,"style":4227},[2064],[1814,5450,5451,5454],{"style":4230},[1814,5452],{"className":5453,"style":2073},[2072],[1814,5455,5457],{"className":5456},[2077,2078,2079,2080],[1814,5458,2958],{"className":5459},[1884,2080],[1814,5461,5380],{"className":5462},[1884],[1814,5464,1967],{"className":5465,"style":2021},[1884,1885],[1814,5467],{"className":5468,"style":1891},[1890],[1814,5470,5189],{"className":5471},[1895],[1814,5473],{"className":5474,"style":1891},[1890],[1814,5476,5478,5481],{"className":5477},[1875],[1814,5479],{"className":5480,"style":2246},[1879],[1814,5482,5387],{"className":5483},[1884],"; latent rates become homogeneous and the marginal count approaches Poisson variance ",[1814,5486,5488,5502],{"className":5487},[1817],[1814,5489,5491],{"className":5490},[1821],[1823,5492,5493],{"xmlns":1825},[1827,5494,5495,5499],{},[1830,5496,5497],{},[1833,5498,2723],{},[1864,5500,5501],{"encoding":1866},"\\mu",[1814,5503,5505],{"className":5504,"ariaHidden":1850},[1871],[1814,5506,5508,5512],{"className":5507},[1875],[1814,5509],{"className":5510,"style":5511},[1879],"height:0.625em;vertical-align:-0.1944em;",[1814,5513,2723],{"className":5514},[1884,1885],[5140,5516,5517],{},"The aggregate mean changes even when every class-specific mean is constant; composition is confounded with within-class trend.",[1793,5519,5520,5521,2425],{},"Next, apply these distributions to ",[5522,5523,5525],"a",{"href":5524},"..\u002F04-reinsurance\u002F","contractually transformed losses",{"title":10,"searchDepth":5527,"depth":5527,"links":5528},2,[5529,5533,5534,5535,5536,5537,5538],{"id":1808,"depth":5527,"text":1809,"children":5530},[5531],{"id":3633,"depth":5532,"text":3634},3,{"id":3825,"depth":5527,"text":3826},{"id":3900,"depth":5527,"text":3901},{"id":4522,"depth":5527,"text":4523},{"id":4841,"depth":5527,"text":4842},{"id":5131,"depth":5527,"text":5132},{"id":5162,"depth":5527,"text":5163},"Represent distinct risk populations and latent states without mistaking flexibility for explanation.","md",{"sidebar":5542},{"order":5543},5,true,{"title":880,"description":5539},"7JJbi6wV8vZvzdpnoWBq2HkVqOAGKXohqSkZXa1UcYY",[5548,5550],{"title":876,"path":877,"stem":878,"description":5549,"children":-1},"Fit the likelihood that generated the observations, then validate the decision-relevant centre and tail.",{"title":884,"path":885,"stem":886,"description":5551,"children":-1},"Translate treaty wording into gross, ceded, and retained random variables before pricing or capital analysis.",1785754735776]