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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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Let ",[1797,1798,1801,1830],"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",[1810,1811,1812,1825],"semantics",{},[1813,1814,1815],"mrow",{},[1816,1817,1818,1822],"msub",{},[1819,1820,1821],"mi",{},"N",[1819,1823,1824],{},"i",[1826,1827,1829],"annotation",{"encoding":1828},"application\u002Fx-tex","N_i",[1797,1831,1835],{"className":1832,"ariaHidden":1834},[1833],"katex-html","true",[1797,1836,1839,1844],{"className":1837},[1838],"base",[1797,1840],{"className":1841,"style":1843},[1842],"strut","height:0.8333em;vertical-align:-0.15em;",[1797,1845,1848,1853],{"className":1846},[1847],"mord",[1797,1849,1821],{"className":1850,"style":1852},[1847,1851],"mathnormal","margin-right:0.109em;",[1797,1854,1857],{"className":1855},[1856],"msupsub",[1797,1858,1862,1894],{"className":1859},[1860,1861],"vlist-t","vlist-t2",[1797,1863,1866,1889],{"className":1864},[1865],"vlist-r",[1797,1867,1871],{"className":1868,"style":1870},[1869],"vlist","height:0.3117em;",[1797,1872,1874,1879],{"style":1873},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[1797,1875],{"className":1876,"style":1878},[1877],"pstrut","height:2.7em;",[1797,1880,1886],{"className":1881},[1882,1883,1884,1885],"sizing","reset-size6","size3","mtight",[1797,1887,1824],{"className":1888},[1847,1851,1885],[1797,1890,1893],{"className":1891},[1892],"vlist-s","​",[1797,1895,1897],{"className":1896},[1865],[1797,1898,1901],{"className":1899,"style":1900},[1869],"height:0.15em;",[1797,1902],{}," be claims from exposure ",[1797,1905,1907,1926],{"className":1906},[1800],[1797,1908,1910],{"className":1909},[1804],[1806,1911,1912],{"xmlns":1808},[1810,1913,1914,1923],{},[1813,1915,1916],{},[1816,1917,1918,1921],{},[1819,1919,1920],{},"e",[1819,1922,1824],{},[1826,1924,1925],{"encoding":1828},"e_i",[1797,1927,1929],{"className":1928,"ariaHidden":1834},[1833],[1797,1930,1932,1936],{"className":1931},[1838],[1797,1933],{"className":1934,"style":1935},[1842],"height:0.5806em;vertical-align:-0.15em;",[1797,1937,1939,1942],{"className":1938},[1847],[1797,1940,1920],{"className":1941},[1847,1851],[1797,1943,1945],{"className":1944},[1856],[1797,1946,1948,1969],{"className":1947},[1860,1861],[1797,1949,1951,1966],{"className":1950},[1865],[1797,1952,1954],{"className":1953,"style":1870},[1869],[1797,1955,1957,1960],{"style":1956},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[1797,1958],{"className":1959,"style":1878},[1877],[1797,1961,1963],{"className":1962},[1882,1883,1884,1885],[1797,1964,1824],{"className":1965},[1847,1851,1885],[1797,1967,1893],{"className":1968},[1892],[1797,1970,1972],{"className":1971},[1865],[1797,1973,1975],{"className":1974,"style":1900},[1869],[1797,1976],{},"—for example, earned vehicle-years—during a stated period.",[1979,1980,1982],"h2",{"id":1981},"_1-begin-with-an-observed-rate","1. Begin with an observed rate",[1984,1985,1986,2009],"table",{},[1987,1988,1989],"thead",{},[1990,1991,1992,1996,2000,2003,2006],"tr",{},[1993,1994,1995],"th",{},"Portfolio",[1993,1997,1999],{"align":1998},"right","Vehicle-years",[1993,2001,2002],{"align":1998},"Claims",[1993,2004,2005],{"align":1998},"Raw count",[1993,2007,2008],{"align":1998},"Claims per vehicle-year",[2010,2011,2012,2030],"tbody",{},[1990,2013,2014,2018,2021,2024,2027],{},[2015,2016,2017],"td",{},"A",[2015,2019,2020],{"align":1998},"10,000",[2015,2022,2023],{"align":1998},"400",[2015,2025,2026],{"align":1998},"lower",[2015,2028,2029],{"align":1998},"0.040",[1990,2031,2032,2035,2038,2041,2044],{},[2015,2033,2034],{},"B",[2015,2036,2037],{"align":1998},"15,000",[2015,2039,2040],{"align":1998},"525",[2015,2042,2043],{"align":1998},"higher",[2015,2045,2046],{"align":1998},"0.035",[1793,2048,2049],{},"Portfolio B has more claims but lower observed frequency. Modelling counts without the exposure denominator reverses the comparison.",[1979,2051,2053],{"id":2052},"_2-poisson-the-benchmark-process","2. Poisson: the benchmark process",[1793,2055,2056,2057,2146],{},"If 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risk ",[1797,2524,2526,2539],{"className":2525},[1800],[1797,2527,2529],{"className":2528},[1804],[1806,2530,2531],{"xmlns":1808},[1810,2532,2533,2537],{},[1813,2534,2535],{},[1819,2536,1824],{},[1826,2538,1824],{"encoding":1828},[1797,2540,2542],{"className":2541,"ariaHidden":1834},[1833],[1797,2543,2545,2549],{"className":2544},[1838],[1797,2546],{"className":2547,"style":2548},[1842],"height:0.6595em;",[1797,2550,1824],{"className":2551},[1847,1851],", write ",[1797,2554,2556,2589],{"className":2555},[1800],[1797,2557,2559],{"className":2558},[1804],[1806,2560,2561],{"xmlns":1808},[1810,2562,2563,2586],{},[1813,2564,2565,2571,2573,2579],{},[1816,2566,2567,2569],{},[1819,2568,2088],{},[1819,2570,1824],{},[2072,2572,2173],{},[1816,2574,2575,2577],{},[1819,2576,1920],{},[1819,2578,1824],{},[1816,2580,2581,2584],{},[1819,2582,2583],{},"r",[1819,2585,1824],{},[1826,2587,2588],{"encoding":1828},"\\lambda_i=e_i r_i",[1797,2590,2592,2648],{"className":2591,"ariaHidden":1834},[1833],[1797,2593,2595,2599,2639,2642,2645],{"className":2594},[1838],[1797,2596],{"className":2597,"style":2598},[1842],"height:0.8444em;vertical-align:-0.15em;",[1797,2600,2602,2605],{"className":2601},[1847],[1797,2603,2088],{"className":2604},[1847,1851],[1797,2606,2608],{"className":2607},[1856],[1797,2609,2611,2631],{"className":2610},[1860,1861],[1797,2612,2614,2628],{"className":2613},[1865],[1797,2615,2617],{"className":2616,"style":1870},[1869],[1797,2618,2619,2622],{"style":1956},[1797,2620],{"className":2621,"style":1878},[1877],[1797,2623,2625],{"className":2624},[1882,1883,1884,1885],[1797,2626,1824],{"className":2627},[1847,1851,1885],[1797,2629,1893],{"className":2630},[1892],[1797,2632,2634],{"className":2633},[1865],[1797,2635,2637],{"className":2636,"style":1900},[1869],[1797,2638],{},[1797,2640],{"className":2641,"style":2112},[2111],[1797,2643,2173],{"className":2644},[2116],[1797,2646],{"className":2647,"style":2112},[2111],[1797,2649,2651,2654,2694],{"className":2650},[1838],[1797,2652],{"className":2653,"style":1935},[1842],[1797,2655,2657,2660],{"className":2656},[1847],[1797,2658,1920],{"className":2659},[1847,1851],[1797,2661,2663],{"className":2662},[1856],[1797,2664,2666,2686],{"className":2665},[1860,1861],[1797,2667,2669,2683],{"className":2668},[1865],[1797,2670,2672],{"className":2671,"style":1870},[1869],[1797,2673,2674,2677],{"style":1956},[1797,2675],{"className":2676,"style":1878},[1877],[1797,2678,2680],{"className":2679},[1882,1883,1884,1885],[1797,2681,1824],{"className":2682},[1847,1851,1885],[1797,2684,1893],{"className":2685},[1892],[1797,2687,2689],{"className":2688},[1865],[1797,2690,2692],{"className":2691,"style":1900},[1869],[1797,2693],{},[1797,2695,2697,2701],{"className":2696},[1847],[1797,2698,2583],{"className":2699,"style":2700},[1847,1851],"margin-right:0.0278em;",[1797,2702,2704],{"className":2703},[1856],[1797,2705,2707,2728],{"className":2706},[1860,1861],[1797,2708,2710,2725],{"className":2709},[1865],[1797,2711,2713],{"className":2712,"style":1870},[1869],[1797,2714,2716,2719],{"style":2715},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[1797,2717],{"className":2718,"style":1878},[1877],[1797,2720,2722],{"className":2721},[1882,1883,1884,1885],[1797,2723,1824],{"className":2724},[1847,1851,1885],[1797,2726,1893],{"className":2727},[1892],[1797,2729,2731],{"className":2730},[1865],[1797,2732,2734],{"className":2733,"style":1900},[1869],[1797,2735],{},", where ",[1797,2738,2740,2758],{"className":2739},[1800],[1797,2741,2743],{"className":2742},[1804],[1806,2744,2745],{"xmlns":1808},[1810,2746,2747,2755],{},[1813,2748,2749],{},[1816,2750,2751,2753],{},[1819,2752,2583],{},[1819,2754,1824],{},[1826,2756,2757],{"encoding":1828},"r_i",[1797,2759,2761],{"className":2760,"ariaHidden":1834},[1833],[1797,2762,2764,2767],{"className":2763},[1838],[1797,2765],{"className":2766,"style":1935},[1842],[1797,2768,2770,2773],{"className":2769},[1847],[1797,2771,2583],{"className":2772,"style":2700},[1847,1851],[1797,2774,2776],{"className":2775},[1856],[1797,2777,2779,2799],{"className":2778},[1860,1861],[1797,2780,2782,2796],{"className":2781},[1865],[1797,2783,2785],{"className":2784,"style":1870},[1869],[1797,2786,2787,2790],{"style":2715},[1797,2788],{"className":2789,"style":1878},[1877],[1797,2791,2793],{"className":2792},[1882,1883,1884,1885],[1797,2794,1824],{"className":2795},[1847,1851,1885],[1797,2797,1893],{"className":2798},[1892],[1797,2800,2802],{"className":2801},[1865],[1797,2803,2805],{"className":2804,"style":1900},[1869],[1797,2806],{}," is expected frequency per unit exposure. A log-link regression becomes",[1797,2809,2811],{"className":2810},[2150],[1797,2812,2814,2884],{"className":2813},[1800],[1797,2815,2817],{"className":2816},[1804],[1806,2818,2819],{"xmlns":1808,"display":2159},[1810,2820,2821,2881],{},[1813,2822,2823,2826,2828,2830,2832,2838,2840,2842,2844,2846,2852,2855,2864,2866,2877,2879],{},[1819,2824,2825],{},"log",[2072,2827,2081],{},[1819,2829,2216],{},[2072,2831,2219],{"stretchy":2084},[1816,2833,2834,2836],{},[1819,2835,1821],{},[1819,2837,1824],{},[2072,2839,2224],{"stretchy":2084},[2072,2841,2173],{},[1819,2843,2825],{},[2072,2845,2081],{},[1816,2847,2848,2850],{},[1819,2849,1920],{},[1819,2851,1824],{},[2072,2853,2854],{},"+",[1816,2856,2857,2860],{},[1819,2858,2859],{},"β",[2861,2862,2863],"mn",{},"0",[2072,2865,2854],{},[2867,2868,2869,2872,2874],"msubsup",{},[1819,2870,2871],{},"x",[1819,2873,1824],{},[1819,2875,2876],{"mathvariant":2077},"⊤",[1819,2878,2859],{},[1819,2880,2244],{"mathvariant":2077},[1826,2882,2883],{"encoding":1828},"\\log E[N_i]=\\log e_i+\\beta_0+x_i^\\top\\beta.",[1797,2885,2887,2962,3028,3086],{"className":2886,"ariaHidden":1834},[1833],[1797,2888,2890,2893,2901,2904,2907,2910,2950,2953,2956,2959],{"className":2889},[1838],[1797,2891],{"className":2892,"style":2126},[1842],[1797,2894,2896,2897],{"className":2895},[2130],"lo",[1797,2898,2900],{"style":2899},"margin-right:0.0139em;","g",[1797,2902],{"className":2903,"style":2454},[2111],[1797,2905,2216],{"className":2906,"style":2458},[1847,1851],[1797,2908,2219],{"className":2909},[2138],[1797,2911,2913,2916],{"className":2912},[1847],[1797,2914,1821],{"className":2915,"style":1852},[1847,1851],[1797,2917,2919],{"className":2918},[1856],[1797,2920,2922,2942],{"className":2921},[1860,1861],[1797,2923,2925,2939],{"className":2924},[1865],[1797,2926,2928],{"className":2927,"style":1870},[1869],[1797,2929,2930,2933],{"style":1873},[1797,2931],{"className":2932,"style":1878},[1877],[1797,2934,2936],{"className":2935},[1882,1883,1884,1885],[1797,2937,1824],{"className":2938},[1847,1851,1885],[1797,2940,1893],{"className":2941},[1892],[1797,2943,2945],{"className":2944},[1865],[1797,2946,2948],{"className":2947,"style":1900},[1869],[1797,2949],{},[1797,2951,2224],{"className":2952},[2145],[1797,2954],{"className":2955,"style":2112},[2111],[1797,2957,2173],{"className":2958},[2116],[1797,2960],{"className":2961,"style":2112},[2111],[1797,2963,2965,2969,2974,2977,3017,3021,3025],{"className":2964},[1838],[1797,2966],{"className":2967,"style":2968},[1842],"height:0.8889em;vertical-align:-0.1944em;",[1797,2970,2896,2972],{"className":2971},[2130],[1797,2973,2900],{"style":2899},[1797,2975],{"className":2976,"style":2454},[2111],[1797,2978,2980,2983],{"className":2979},[1847],[1797,2981,1920],{"className":2982},[1847,1851],[1797,2984,2986],{"className":2985},[1856],[1797,2987,2989,3009],{"className":2988},[1860,1861],[1797,2990,2992,3006],{"className":2991},[1865],[1797,2993,2995],{"className":2994,"style":1870},[1869],[1797,2996,2997,3000],{"style":1956},[1797,2998],{"className":2999,"style":1878},[1877],[1797,3001,3003],{"className":3002},[1882,1883,1884,1885],[1797,3004,1824],{"className":3005},[1847,1851,1885],[1797,3007,1893],{"className":3008},[1892],[1797,3010,3012],{"className":3011},[1865],[1797,3013,3015],{"className":3014,"style":1900},[1869],[1797,3016],{},[1797,3018],{"className":3019,"style":3020},[2111],"margin-right:0.2222em;",[1797,3022,2854],{"className":3023},[3024],"mbin",[1797,3026],{"className":3027,"style":3020},[2111],[1797,3029,3031,3034,3077,3080,3083],{"className":3030},[1838],[1797,3032],{"className":3033,"style":2968},[1842],[1797,3035,3037,3041],{"className":3036},[1847],[1797,3038,2859],{"className":3039,"style":3040},[1847,1851],"margin-right:0.0528em;",[1797,3042,3044],{"className":3043},[1856],[1797,3045,3047,3069],{"className":3046},[1860,1861],[1797,3048,3050,3066],{"className":3049},[1865],[1797,3051,3054],{"className":3052,"style":3053},[1869],"height:0.3011em;",[1797,3055,3057,3060],{"style":3056},"top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;",[1797,3058],{"className":3059,"style":1878},[1877],[1797,3061,3063],{"className":3062},[1882,1883,1884,1885],[1797,3064,2863],{"className":3065},[1847,1885],[1797,3067,1893],{"className":3068},[1892],[1797,3070,3072],{"className":3071},[1865],[1797,3073,3075],{"className":3074,"style":1900},[1869],[1797,3076],{},[1797,3078],{"className":3079,"style":3020},[2111],[1797,3081,2854],{"className":3082},[3024],[1797,3084],{"className":3085,"style":3020},[2111],[1797,3087,3089,3093,3146,3149],{"className":3088},[1838],[1797,3090],{"className":3091,"style":3092},[1842],"height:1.1461em;vertical-align:-0.247em;",[1797,3094,3096,3099],{"className":3095},[1847],[1797,3097,2871],{"className":3098},[1847,1851],[1797,3100,3102],{"className":3101},[1856],[1797,3103,3105,3137],{"className":3104},[1860,1861],[1797,3106,3108,3134],{"className":3107},[1865],[1797,3109,3111,3123],{"className":3110,"style":2322},[1869],[1797,3112,3114,3117],{"style":3113},"top:-2.453em;margin-left:0em;margin-right:0.05em;",[1797,3115],{"className":3116,"style":1878},[1877],[1797,3118,3120],{"className":3119},[1882,1883,1884,1885],[1797,3121,1824],{"className":3122},[1847,1851,1885],[1797,3124,3125,3128],{"style":2325},[1797,3126],{"className":3127,"style":1878},[1877],[1797,3129,3131],{"className":3130},[1882,1883,1884,1885],[1797,3132,2876],{"className":3133},[1847,1885],[1797,3135,1893],{"className":3136},[1892],[1797,3138,3140],{"className":3139},[1865],[1797,3141,3144],{"className":3142,"style":3143},[1869],"height:0.247em;",[1797,3145],{},[1797,3147,2859],{"className":3148,"style":3040},[1847,1851],[1797,3150,2244],{"className":3151},[1847],[1793,3153,3154,3236,3237,3241,3242,3272,3273,3360],{},[1797,3155,3157,3179],{"className":3156},[1800],[1797,3158,3160],{"className":3159},[1804],[1806,3161,3162],{"xmlns":1808},[1810,3163,3164,3176],{},[1813,3165,3166,3168,3170],{},[1819,3167,2825],{},[2072,3169,2081],{},[1816,3171,3172,3174],{},[1819,3173,1920],{},[1819,3175,1824],{},[1826,3177,3178],{"encoding":1828},"\\log e_i",[1797,3180,3182],{"className":3181,"ariaHidden":1834},[1833],[1797,3183,3185,3188,3193,3196],{"className":3184},[1838],[1797,3186],{"className":3187,"style":2968},[1842],[1797,3189,2896,3191],{"className":3190},[2130],[1797,3192,2900],{"style":2899},[1797,3194],{"className":3195,"style":2454},[2111],[1797,3197,3199,3202],{"className":3198},[1847],[1797,3200,1920],{"className":3201},[1847,1851],[1797,3203,3205],{"className":3204},[1856],[1797,3206,3208,3228],{"className":3207},[1860,1861],[1797,3209,3211,3225],{"className":3210},[1865],[1797,3212,3214],{"className":3213,"style":1870},[1869],[1797,3215,3216,3219],{"style":1956},[1797,3217],{"className":3218,"style":1878},[1877],[1797,3220,3222],{"className":3221},[1882,1883,1884,1885],[1797,3223,1824],{"className":3224},[1847,1851,1885],[1797,3226,1893],{"className":3227},[1892],[1797,3229,3231],{"className":3230},[1865],[1797,3232,3234],{"className":3233,"style":1900},[1869],[1797,3235],{}," is an ",[3238,3239,3240],"strong",{},"offset",": its coefficient is fixed at one. If a rating factor has coefficient ",[1797,3243,3245,3259],{"className":3244},[1800],[1797,3246,3248],{"className":3247},[1804],[1806,3249,3250],{"xmlns":1808},[1810,3251,3252,3257],{},[1813,3253,3254],{},[2861,3255,3256],{},"0.20",[1826,3258,3256],{"encoding":1828},[1797,3260,3262],{"className":3261,"ariaHidden":1834},[1833],[1797,3263,3265,3269],{"className":3264},[1838],[1797,3266],{"className":3267,"style":3268},[1842],"height:0.6444em;",[1797,3270,3256],{"className":3271},[1847],", its multiplicative effect on expected frequency is ",[1797,3274,3276,3300],{"className":3275},[1800],[1797,3277,3279],{"className":3278},[1804],[1806,3280,3281],{"xmlns":1808},[1810,3282,3283,3297],{},[1813,3284,3285,3291,3294],{},[2182,3286,3287,3289],{},[1819,3288,1920],{},[2861,3290,3256],{},[2072,3292,3293],{},"≈",[2861,3295,3296],{},"1.22",[1826,3298,3299],{"encoding":1828},"e^{0.20}\\approx1.22",[1797,3301,3303,3351],{"className":3302,"ariaHidden":1834},[1833],[1797,3304,3306,3310,3342,3345,3348],{"className":3305},[1838],[1797,3307],{"className":3308,"style":3309},[1842],"height:0.8141em;",[1797,3311,3313,3316],{"className":3312},[1847],[1797,3314,1920],{"className":3315},[1847,1851],[1797,3317,3319],{"className":3318},[1856],[1797,3320,3322],{"className":3321},[1860],[1797,3323,3325],{"className":3324},[1865],[1797,3326,3328],{"className":3327,"style":3309},[1869],[1797,3329,3330,3333],{"style":2418},[1797,3331],{"className":3332,"style":1878},[1877],[1797,3334,3336],{"className":3335},[1882,1883,1884,1885],[1797,3337,3339],{"className":3338},[1847,1885],[1797,3340,3256],{"className":3341},[1847,1885],[1797,3343],{"className":3344,"style":2112},[2111],[1797,3346,3293],{"className":3347},[2116],[1797,3349],{"className":3350,"style":2112},[2111],[1797,3352,3354,3357],{"className":3353},[1838],[1797,3355],{"className":3356,"style":3268},[1842],[1797,3358,3296],{"className":3359},[1847],", holding other variables and exposure fixed.",[3362,3363,3365],"h3",{"id":3364},"why-poisson-appears","Why Poisson appears",[1793,3367,3368],{},"Poisson is plausible when many opportunities each have small event probability and events are approximately independent over a fixed exposure. These are process claims to investigate, not consequences of integer-valued data.",[1979,3370,3372],{"id":3371},"_3-binomial-fixed-opportunities","3. Binomial: fixed opportunities",[1793,3374,3375,3376,3406,3407,3436],{},"If exactly ",[1797,3377,3379,3393],{"className":3378},[1800],[1797,3380,3382],{"className":3381},[1804],[1806,3383,3384],{"xmlns":1808},[1810,3385,3386,3391],{},[1813,3387,3388],{},[1819,3389,3390],{},"m",[1826,3392,3390],{"encoding":1828},[1797,3394,3396],{"className":3395,"ariaHidden":1834},[1833],[1797,3397,3399,3403],{"className":3398},[1838],[1797,3400],{"className":3401,"style":3402},[1842],"height:0.4306em;",[1797,3404,3390],{"className":3405},[1847,1851]," policies can each produce one event with probability ",[1797,3408,3410,3423],{"className":3409},[1800],[1797,3411,3413],{"className":3412},[1804],[1806,3414,3415],{"xmlns":1808},[1810,3416,3417,3421],{},[1813,3418,3419],{},[1819,3420,1793],{},[1826,3422,1793],{"encoding":1828},[1797,3424,3426],{"className":3425,"ariaHidden":1834},[1833],[1797,3427,3429,3433],{"className":3428},[1838],[1797,3430],{"className":3431,"style":3432},[1842],"height:0.625em;vertical-align:-0.1944em;",[1797,3434,1793],{"className":3435},[1847,1851]," independently,",[1797,3438,3440],{"className":3439},[2150],[1797,3441,3443,3525],{"className":3442},[1800],[1797,3444,3446],{"className":3445},[1804],[1806,3447,3448],{"xmlns":1808,"display":2159},[1810,3449,3450,3522],{},[1813,3451,3452,3454,3456,3459,3461,3463,3465,3467,3469,3471,3473,3475,3477,3479,3481,3483,3485,3487,3489,3491,3493,3495,3497,3499,3501,3503,3505,3507,3509,3511,3514,3516,3518,3520],{},[1819,3453,1821],{},[2072,3455,2074],{},[1819,3457,3458],{"mathvariant":2077},"Binomial",[2072,3460,2081],{},[2072,3462,2085],{"stretchy":2084},[1819,3464,3390],{},[2072,3466,2146],{"separator":1834},[1819,3468,1793],{},[2072,3470,2091],{"stretchy":2084},[2072,3472,2146],{"separator":1834},[2111,3474],{"width":2213},[1819,3476,2216],{},[2072,3478,2219],{"stretchy":2084},[1819,3480,1821],{},[2072,3482,2224],{"stretchy":2084},[2072,3484,2173],{},[1819,3486,3390],{},[1819,3488,1793],{},[2072,3490,2146],{"separator":1834},[2111,3492],{"width":2213},[1819,3494,2229],{"mathvariant":2077},[2072,3496,2081],{},[2072,3498,2085],{"stretchy":2084},[1819,3500,1821],{},[2072,3502,2091],{"stretchy":2084},[2072,3504,2173],{},[1819,3506,3390],{},[1819,3508,1793],{},[2072,3510,2085],{"stretchy":2084},[2861,3512,3513],{},"1",[2072,3515,2190],{},[1819,3517,1793],{},[2072,3519,2091],{"stretchy":2084},[1819,3521,2244],{"mathvariant":2077},[1826,3523,3524],{"encoding":1828},"N\\sim\\operatorname{Binomial}(m,p),\n\\qquad E[N]=mp,\n\\qquad \\operatorname{Var}(N)=mp(1-p).",[1797,3526,3528,3546,3606,3651,3678],{"className":3527,"ariaHidden":1834},[1833],[1797,3529,3531,3534,3537,3540,3543],{"className":3530},[1838],[1797,3532],{"className":3533,"style":2104},[1842],[1797,3535,1821],{"className":3536,"style":1852},[1847,1851],[1797,3538],{"className":3539,"style":2112},[2111],[1797,3541,2074],{"className":3542},[2116],[1797,3544],{"className":3545,"style":2112},[2111],[1797,3547,3549,3552,3558,3561,3564,3567,3570,3573,3576,3579,3582,3585,3588,3591,3594,3597,3600,3603],{"className":3548},[1838],[1797,3550],{"className":3551,"style":2126},[1842],[1797,3553,3555],{"className":3554},[2130],[1797,3556,3458],{"className":3557},[1847,2134],[1797,3559,2085],{"className":3560},[2138],[1797,3562,3390],{"className":3563},[1847,1851],[1797,3565,2146],{"className":3566},[2446],[1797,3568],{"className":3569,"style":2454},[2111],[1797,3571,1793],{"className":3572},[1847,1851],[1797,3574,2091],{"className":3575},[2145],[1797,3577,2146],{"className":3578},[2446],[1797,3580],{"className":3581,"style":2450},[2111],[1797,3583],{"className":3584,"style":2454},[2111],[1797,3586,2216],{"className":3587,"style":2458},[1847,1851],[1797,3589,2219],{"className":3590},[2138],[1797,3592,1821],{"className":3593,"style":1852},[1847,1851],[1797,3595,2224],{"className":3596},[2145],[1797,3598],{"className":3599,"style":2112},[2111],[1797,3601,2173],{"className":3602},[2116],[1797,3604],{"className":3605,"style":2112},[2111],[1797,3607,3609,3612,3615,3618,3621,3624,3627,3633,3636,3639,3642,3645,3648],{"className":3608},[1838],[1797,3610],{"className":3611,"style":2126},[1842],[1797,3613,3390],{"className":3614},[1847,1851],[1797,3616,1793],{"className":3617},[1847,1851],[1797,3619,2146],{"className":3620},[2446],[1797,3622],{"className":3623,"style":2450},[2111],[1797,3625],{"className":3626,"style":2454},[2111],[1797,3628,3630],{"className":3629},[2130],[1797,3631,2229],{"className":3632},[1847,2134],[1797,3634,2085],{"className":3635},[2138],[1797,3637,1821],{"className":3638,"style":1852},[1847,1851],[1797,3640,2091],{"className":3641},[2145],[1797,3643],{"className":3644,"style":2112},[2111],[1797,3646,2173],{"className":3647},[2116],[1797,3649],{"className":3650,"style":2112},[2111],[1797,3652,3654,3657,3660,3663,3666,3669,3672,3675],{"className":3653},[1838],[1797,3655],{"className":3656,"style":2126},[1842],[1797,3658,3390],{"className":3659},[1847,1851],[1797,3661,1793],{"className":3662},[1847,1851],[1797,3664,2085],{"className":3665},[2138],[1797,3667,3513],{"className":3668},[1847],[1797,3670],{"className":3671,"style":3020},[2111],[1797,3673,2190],{"className":3674},[3024],[1797,3676],{"className":3677,"style":3020},[2111],[1797,3679,3681,3684,3687,3690],{"className":3680},[1838],[1797,3682],{"className":3683,"style":2126},[1842],[1797,3685,1793],{"className":3686},[1847,1851],[1797,3688,2091],{"className":3689},[2145],[1797,3691,2244],{"className":3692},[1847],[1793,3694,3695],{},"Binomial is appropriate for “at least one claim” per policy. It is not appropriate when one policy can generate several claims unless the response is deliberately reduced to a binary indicator.",[1979,3697,3699],{"id":3698},"_4-negative-binomial-extra-poisson-variation","4. Negative binomial: extra-Poisson variation",[1793,3701,3702],{},"A useful parameterisation has",[1797,3704,3706],{"className":3705},[2150],[1797,3707,3709,3794],{"className":3708},[1800],[1797,3710,3712],{"className":3711},[1804],[1806,3713,3714],{"xmlns":1808,"display":2159},[1810,3715,3716,3791],{},[1813,3717,3718,3720,3722,3728,3730,3732,3739,3741,3743,3745,3747,3749,3755,3757,3759,3765,3767,3770,3779,3781,3783,3785,3788],{},[1819,3719,2216],{},[2072,3721,2219],{"stretchy":2084},[1816,3723,3724,3726],{},[1819,3725,1821],{},[1819,3727,1824],{},[2072,3729,2224],{"stretchy":2084},[2072,3731,2173],{},[1816,3733,3734,3737],{},[1819,3735,3736],{},"μ",[1819,3738,1824],{},[2072,3740,2146],{"separator":1834},[2111,3742],{"width":2213},[1819,3744,2229],{"mathvariant":2077},[2072,3746,2081],{},[2072,3748,2085],{"stretchy":2084},[1816,3750,3751,3753],{},[1819,3752,1821],{},[1819,3754,1824],{},[2072,3756,2091],{"stretchy":2084},[2072,3758,2173],{},[1816,3760,3761,3763],{},[1819,3762,3736],{},[1819,3764,1824],{},[2072,3766,2854],{},[1819,3768,3769],{},"α",[2867,3771,3772,3774,3776],{},[1819,3773,3736],{},[1819,3775,1824],{},[2861,3777,3778],{},"2",[2072,3780,2146],{"separator":1834},[2111,3782],{"width":2213},[1819,3784,3769],{},[2072,3786,3787],{},">",[2861,3789,3790],{},"0.",[1826,3792,3793],{"encoding":1828},"E[N_i]=\\mu_i,\n\\qquad\n\\operatorname{Var}(N_i)=\\mu_i+\\alpha\\mu_i^2,\n\\qquad \\alpha>0.",[1797,3795,3797,3861,3977,4033,4117],{"className":3796,"ariaHidden":1834},[1833],[1797,3798,3800,3803,3806,3809,3849,3852,3855,3858],{"className":3799},[1838],[1797,3801],{"className":3802,"style":2126},[1842],[1797,3804,2216],{"className":3805,"style":2458},[1847,1851],[1797,3807,2219],{"className":3808},[2138],[1797,3810,3812,3815],{"className":3811},[1847],[1797,3813,1821],{"className":3814,"style":1852},[1847,1851],[1797,3816,3818],{"className":3817},[1856],[1797,3819,3821,3841],{"className":3820},[1860,1861],[1797,3822,3824,3838],{"className":3823},[1865],[1797,3825,3827],{"className":3826,"style":1870},[1869],[1797,3828,3829,3832],{"style":1873},[1797,3830],{"className":3831,"style":1878},[1877],[1797,3833,3835],{"className":3834},[1882,1883,1884,1885],[1797,3836,1824],{"className":3837},[1847,1851,1885],[1797,3839,1893],{"className":3840},[1892],[1797,3842,3844],{"className":3843},[1865],[1797,3845,3847],{"className":3846,"style":1900},[1869],[1797,3848],{},[1797,3850,2224],{"className":3851},[2145],[1797,3853],{"className":3854,"style":2112},[2111],[1797,3856,2173],{"className":3857},[2116],[1797,3859],{"className":3860,"style":2112},[2111],[1797,3862,3864,3867,3907,3910,3913,3916,3922,3925,3965,3968,3971,3974],{"className":3863},[1838],[1797,3865],{"className":3866,"style":2126},[1842],[1797,3868,3870,3873],{"className":3869},[1847],[1797,3871,3736],{"className":3872},[1847,1851],[1797,3874,3876],{"className":3875},[1856],[1797,3877,3879,3899],{"className":3878},[1860,1861],[1797,3880,3882,3896],{"className":3881},[1865],[1797,3883,3885],{"className":3884,"style":1870},[1869],[1797,3886,3887,3890],{"style":1956},[1797,3888],{"className":3889,"style":1878},[1877],[1797,3891,3893],{"className":3892},[1882,1883,1884,1885],[1797,3894,1824],{"className":3895},[1847,1851,1885],[1797,3897,1893],{"className":3898},[1892],[1797,3900,3902],{"className":3901},[1865],[1797,3903,3905],{"className":3904,"style":1900},[1869],[1797,3906],{},[1797,3908,2146],{"className":3909},[2446],[1797,3911],{"className":3912,"style":2450},[2111],[1797,3914],{"className":3915,"style":2454},[2111],[1797,3917,3919],{"className":3918},[2130],[1797,3920,2229],{"className":3921},[1847,2134],[1797,3923,2085],{"className":3924},[2138],[1797,3926,3928,3931],{"className":3927},[1847],[1797,3929,1821],{"className":3930,"style":1852},[1847,1851],[1797,3932,3934],{"className":3933},[1856],[1797,3935,3937,3957],{"className":3936},[1860,1861],[1797,3938,3940,3954],{"className":3939},[1865],[1797,3941,3943],{"className":3942,"style":1870},[1869],[1797,3944,3945,3948],{"style":1873},[1797,3946],{"className":3947,"style":1878},[1877],[1797,3949,3951],{"className":3950},[1882,1883,1884,1885],[1797,3952,1824],{"className":3953},[1847,1851,1885],[1797,3955,1893],{"className":3956},[1892],[1797,3958,3960],{"className":3959},[1865],[1797,3961,3963],{"className":3962,"style":1900},[1869],[1797,3964],{},[1797,3966,2091],{"className":3967},[2145],[1797,3969],{"className":3970,"style":2112},[2111],[1797,3972,2173],{"className":3973},[2116],[1797,3975],{"className":3976,"style":2112},[2111],[1797,3978,3980,3984,4024,4027,4030],{"className":3979},[1838],[1797,3981],{"className":3982,"style":3983},[1842],"height:0.7778em;vertical-align:-0.1944em;",[1797,3985,3987,3990],{"className":3986},[1847],[1797,3988,3736],{"className":3989},[1847,1851],[1797,3991,3993],{"className":3992},[1856],[1797,3994,3996,4016],{"className":3995},[1860,1861],[1797,3997,3999,4013],{"className":3998},[1865],[1797,4000,4002],{"className":4001,"style":1870},[1869],[1797,4003,4004,4007],{"style":1956},[1797,4005],{"className":4006,"style":1878},[1877],[1797,4008,4010],{"className":4009},[1882,1883,1884,1885],[1797,4011,1824],{"className":4012},[1847,1851,1885],[1797,4014,1893],{"className":4015},[1892],[1797,4017,4019],{"className":4018},[1865],[1797,4020,4022],{"className":4021,"style":1900},[1869],[1797,4023],{},[1797,4025],{"className":4026,"style":3020},[2111],[1797,4028,2854],{"className":4029},[3024],[1797,4031],{"className":4032,"style":3020},[2111],[1797,4034,4036,4040,4044,4096,4099,4102,4105,4108,4111,4114],{"className":4035},[1838],[1797,4037],{"className":4038,"style":4039},[1842],"height:1.1111em;vertical-align:-0.247em;",[1797,4041,3769],{"className":4042,"style":4043},[1847,1851],"margin-right:0.0037em;",[1797,4045,4047,4050],{"className":4046},[1847],[1797,4048,3736],{"className":4049},[1847,1851],[1797,4051,4053],{"className":4052},[1856],[1797,4054,4056,4088],{"className":4055},[1860,1861],[1797,4057,4059,4085],{"className":4058},[1865],[1797,4060,4063,4074],{"className":4061,"style":4062},[1869],"height:0.8641em;",[1797,4064,4065,4068],{"style":3113},[1797,4066],{"className":4067,"style":1878},[1877],[1797,4069,4071],{"className":4070},[1882,1883,1884,1885],[1797,4072,1824],{"className":4073},[1847,1851,1885],[1797,4075,4076,4079],{"style":2325},[1797,4077],{"className":4078,"style":1878},[1877],[1797,4080,4082],{"className":4081},[1882,1883,1884,1885],[1797,4083,3778],{"className":4084},[1847,1885],[1797,4086,1893],{"className":4087},[1892],[1797,4089,4091],{"className":4090},[1865],[1797,4092,4094],{"className":4093,"style":3143},[1869],[1797,4095],{},[1797,4097,2146],{"className":4098},[2446],[1797,4100],{"className":4101,"style":2450},[2111],[1797,4103],{"className":4104,"style":2454},[2111],[1797,4106,3769],{"className":4107,"style":4043},[1847,1851],[1797,4109],{"className":4110,"style":2112},[2111],[1797,4112,3787],{"className":4113},[2116],[1797,4115],{"className":4116,"style":2112},[2111],[1797,4118,4120,4123],{"className":4119},[1838],[1797,4121],{"className":4122,"style":3268},[1842],[1797,4124,3790],{"className":4125},[1847],[1793,4127,4128],{},"It can arise from Poisson counts with a Gamma-distributed latent rate. Different policyholders then have different unobserved propensities, even after measured rating factors are included.",[3362,4130,4132],{"id":4131},"diagnose-do-not-merely-switch-families","Diagnose, do not merely switch families",[1793,4134,4135],{},"Overdispersion—sample or residual variance exceeding the Poisson mean—can arise from:",[4137,4138,4139,4143,4146,4149,4152,4155],"ul",{},[4140,4141,4142],"li",{},"unobserved heterogeneity;",[4140,4144,4145],{},"common weather or event shocks;",[4140,4147,4148],{},"omitted seasonality or trend;",[4140,4150,4151],{},"excess zeros;",[4140,4153,4154],{},"exposure errors;",[4140,4156,4157],{},"serial or spatial dependence.",[1793,4159,4160],{},"Negative binomial handles one variance pattern but does not identify which mechanism caused it.",[1979,4162,4164],{"id":4163},"_5-zero-inflated-and-hurdle-models","5. Zero-inflated and hurdle models",[1793,4166,4167],{},"Two distinct questions can generate excess zeros:",[4137,4169,4170,4176],{},[4140,4171,4172,4175],{},[3238,4173,4174],{},"Zero-inflated model:"," some observations are in a structural-zero state; the remaining state can also produce zeros.",[4140,4177,4178,4181],{},[3238,4179,4180],{},"Hurdle model:"," one process determines zero versus positive; a zero-truncated count model determines positive counts.",[1793,4183,4184],{},"Use them only when the zero mechanism has operational meaning. A flexible model can fit zeros while giving misleading interventions.",[1979,4186,4188],{"id":4187},"_6-time-and-dependence","6. Time and dependence",[1793,4190,4191],{},"For monthly or regional counts, condition on known structure:",[1797,4193,4195],{"className":4194},[2150],[1797,4196,4198,4293],{"className":4197},[1800],[1797,4199,4201],{"className":4200},[1804],[1806,4202,4203],{"xmlns":1808,"display":2159},[1810,4204,4205,4290],{},[1813,4206,4207,4209,4211,4213,4215,4226,4228,4230,4232,4234,4244,4246,4252,4254,4266,4268,4270,4273,4275,4277,4279,4281,4288],{},[1819,4208,2825],{},[2072,4210,2081],{},[1819,4212,2216],{},[2072,4214,2219],{"stretchy":2084},[1816,4216,4217,4219],{},[1819,4218,1821],{},[1813,4220,4221,4223],{},[1819,4222,1824],{},[1819,4224,4225],{},"t",[2072,4227,2224],{"stretchy":2084},[2072,4229,2173],{},[1819,4231,2825],{},[2072,4233,2081],{},[1816,4235,4236,4238],{},[1819,4237,1920],{},[1813,4239,4240,4242],{},[1819,4241,1824],{},[1819,4243,4225],{},[2072,4245,2854],{},[1816,4247,4248,4250],{},[1819,4249,2859],{},[2861,4251,2863],{},[2072,4253,2854],{},[2867,4255,4256,4258,4264],{},[1819,4257,2871],{},[1813,4259,4260,4262],{},[1819,4261,1824],{},[1819,4263,4225],{},[1819,4265,2876],{"mathvariant":2077},[1819,4267,2859],{},[2072,4269,2854],{},[1819,4271,4272],{},"s",[2072,4274,2085],{"stretchy":2084},[1819,4276,4225],{},[2072,4278,2091],{"stretchy":2084},[2072,4280,2854],{},[1816,4282,4283,4286],{},[1819,4284,4285],{},"b",[1819,4287,1824],{},[2072,4289,2146],{"separator":1834},[1826,4291,4292],{"encoding":1828},"\\log E[N_{it}]\n=\\log e_{it}+\\beta_0+x_{it}^{\\top}\\beta+s(t)+b_i,",[1797,4294,4296,4374,4443,4498,4576,4603],{"className":4295,"ariaHidden":1834},[1833],[1797,4297,4299,4302,4307,4310,4313,4316,4362,4365,4368,4371],{"className":4298},[1838],[1797,4300],{"className":4301,"style":2126},[1842],[1797,4303,2896,4305],{"className":4304},[2130],[1797,4306,2900],{"style":2899},[1797,4308],{"className":4309,"style":2454},[2111],[1797,4311,2216],{"className":4312,"style":2458},[1847,1851],[1797,4314,2219],{"className":4315},[2138],[1797,4317,4319,4322],{"className":4318},[1847],[1797,4320,1821],{"className":4321,"style":1852},[1847,1851],[1797,4323,4325],{"className":4324},[1856],[1797,4326,4328,4354],{"className":4327},[1860,1861],[1797,4329,4331,4351],{"className":4330},[1865],[1797,4332,4334],{"className":4333,"style":1870},[1869],[1797,4335,4336,4339],{"style":1873},[1797,4337],{"className":4338,"style":1878},[1877],[1797,4340,4342],{"className":4341},[1882,1883,1884,1885],[1797,4343,4345,4348],{"className":4344},[1847,1885],[1797,4346,1824],{"className":4347},[1847,1851,1885],[1797,4349,4225],{"className":4350},[1847,1851,1885],[1797,4352,1893],{"className":4353},[1892],[1797,4355,4357],{"className":4356},[1865],[1797,4358,4360],{"className":4359,"style":1900},[1869],[1797,4361],{},[1797,4363,2224],{"className":4364},[2145],[1797,4366],{"className":4367,"style":2112},[2111],[1797,4369,2173],{"className":4370},[2116],[1797,4372],{"className":4373,"style":2112},[2111],[1797,4375,4377,4380,4385,4388,4434,4437,4440],{"className":4376},[1838],[1797,4378],{"className":4379,"style":2968},[1842],[1797,4381,2896,4383],{"className":4382},[2130],[1797,4384,2900],{"style":2899},[1797,4386],{"className":4387,"style":2454},[2111],[1797,4389,4391,4394],{"className":4390},[1847],[1797,4392,1920],{"className":4393},[1847,1851],[1797,4395,4397],{"className":4396},[1856],[1797,4398,4400,4426],{"className":4399},[1860,1861],[1797,4401,4403,4423],{"className":4402},[1865],[1797,4404,4406],{"className":4405,"style":1870},[1869],[1797,4407,4408,4411],{"style":1956},[1797,4409],{"className":4410,"style":1878},[1877],[1797,4412,4414],{"className":4413},[1882,1883,1884,1885],[1797,4415,4417,4420],{"className":4416},[1847,1885],[1797,4418,1824],{"className":4419},[1847,1851,1885],[1797,4421,4225],{"className":4422},[1847,1851,1885],[1797,4424,1893],{"className":4425},[1892],[1797,4427,4429],{"className":4428},[1865],[1797,4430,4432],{"className":4431,"style":1900},[1869],[1797,4433],{},[1797,4435],{"className":4436,"style":3020},[2111],[1797,4438,2854],{"className":4439},[3024],[1797,4441],{"className":4442,"style":3020},[2111],[1797,4444,4446,4449,4489,4492,4495],{"className":4445},[1838],[1797,4447],{"className":4448,"style":2968},[1842],[1797,4450,4452,4455],{"className":4451},[1847],[1797,4453,2859],{"className":4454,"style":3040},[1847,1851],[1797,4456,4458],{"className":4457},[1856],[1797,4459,4461,4481],{"className":4460},[1860,1861],[1797,4462,4464,4478],{"className":4463},[1865],[1797,4465,4467],{"className":4466,"style":3053},[1869],[1797,4468,4469,4472],{"style":3056},[1797,4470],{"className":4471,"style":1878},[1877],[1797,4473,4475],{"className":4474},[1882,1883,1884,1885],[1797,4476,2863],{"className":4477},[1847,1885],[1797,4479,1893],{"className":4480},[1892],[1797,4482,4484],{"className":4483},[1865],[1797,4485,4487],{"className":4486,"style":1900},[1869],[1797,4488],{},[1797,4490],{"className":4491,"style":3020},[2111],[1797,4493,2854],{"className":4494},[3024],[1797,4496],{"className":4497,"style":3020},[2111],[1797,4499,4501,4504,4564,4567,4570,4573],{"className":4500},[1838],[1797,4502],{"className":4503,"style":3092},[1842],[1797,4505,4507,4510],{"className":4506},[1847],[1797,4508,2871],{"className":4509},[1847,1851],[1797,4511,4513],{"className":4512},[1856],[1797,4514,4516,4556],{"className":4515},[1860,1861],[1797,4517,4519,4553],{"className":4518},[1865],[1797,4520,4522,4539],{"className":4521,"style":2322},[1869],[1797,4523,4524,4527],{"style":3113},[1797,4525],{"className":4526,"style":1878},[1877],[1797,4528,4530],{"className":4529},[1882,1883,1884,1885],[1797,4531,4533,4536],{"className":4532},[1847,1885],[1797,4534,1824],{"className":4535},[1847,1851,1885],[1797,4537,4225],{"className":4538},[1847,1851,1885],[1797,4540,4541,4544],{"style":2325},[1797,4542],{"className":4543,"style":1878},[1877],[1797,4545,4547],{"className":4546},[1882,1883,1884,1885],[1797,4548,4550],{"className":4549},[1847,1885],[1797,4551,2876],{"className":4552},[1847,1885],[1797,4554,1893],{"className":4555},[1892],[1797,4557,4559],{"className":4558},[1865],[1797,4560,4562],{"className":4561,"style":3143},[1869],[1797,4563],{},[1797,4565,2859],{"className":4566,"style":3040},[1847,1851],[1797,4568],{"className":4569,"style":3020},[2111],[1797,4571,2854],{"className":4572},[3024],[1797,4574],{"className":4575,"style":3020},[2111],[1797,4577,4579,4582,4585,4588,4591,4594,4597,4600],{"className":4578},[1838],[1797,4580],{"className":4581,"style":2126},[1842],[1797,4583,4272],{"className":4584},[1847,1851],[1797,4586,2085],{"className":4587},[2138],[1797,4589,4225],{"className":4590},[1847,1851],[1797,4592,2091],{"className":4593},[2145],[1797,4595],{"className":4596,"style":3020},[2111],[1797,4598,2854],{"className":4599},[3024],[1797,4601],{"className":4602,"style":3020},[2111],[1797,4604,4606,4609,4649],{"className":4605},[1838],[1797,4607],{"className":4608,"style":2968},[1842],[1797,4610,4612,4615],{"className":4611},[1847],[1797,4613,4285],{"className":4614},[1847,1851],[1797,4616,4618],{"className":4617},[1856],[1797,4619,4621,4641],{"className":4620},[1860,1861],[1797,4622,4624,4638],{"className":4623},[1865],[1797,4625,4627],{"className":4626,"style":1870},[1869],[1797,4628,4629,4632],{"style":1956},[1797,4630],{"className":4631,"style":1878},[1877],[1797,4633,4635],{"className":4634},[1882,1883,1884,1885],[1797,4636,1824],{"className":4637},[1847,1851,1885],[1797,4639,1893],{"className":4640},[1892],[1797,4642,4644],{"className":4643},[1865],[1797,4645,4647],{"className":4646,"style":1900},[1869],[1797,4648],{},[1797,4650,2146],{"className":4651},[2446],[1793,4653,4654,4655,4699,4700,4770],{},"where ",[1797,4656,4658,4678],{"className":4657},[1800],[1797,4659,4661],{"className":4660},[1804],[1806,4662,4663],{"xmlns":1808},[1810,4664,4665,4675],{},[1813,4666,4667,4669,4671,4673],{},[1819,4668,4272],{},[2072,4670,2085],{"stretchy":2084},[1819,4672,4225],{},[2072,4674,2091],{"stretchy":2084},[1826,4676,4677],{"encoding":1828},"s(t)",[1797,4679,4681],{"className":4680,"ariaHidden":1834},[1833],[1797,4682,4684,4687,4690,4693,4696],{"className":4683},[1838],[1797,4685],{"className":4686,"style":2126},[1842],[1797,4688,4272],{"className":4689},[1847,1851],[1797,4691,2085],{"className":4692},[2138],[1797,4694,4225],{"className":4695},[1847,1851],[1797,4697,2091],{"className":4698},[2145]," can represent seasonality\u002Ftrend and ",[1797,4701,4703,4721],{"className":4702},[1800],[1797,4704,4706],{"className":4705},[1804],[1806,4707,4708],{"xmlns":1808},[1810,4709,4710,4718],{},[1813,4711,4712],{},[1816,4713,4714,4716],{},[1819,4715,4285],{},[1819,4717,1824],{},[1826,4719,4720],{"encoding":1828},"b_i",[1797,4722,4724],{"className":4723,"ariaHidden":1834},[1833],[1797,4725,4727,4730],{"className":4726},[1838],[1797,4728],{"className":4729,"style":2598},[1842],[1797,4731,4733,4736],{"className":4732},[1847],[1797,4734,4285],{"className":4735},[1847,1851],[1797,4737,4739],{"className":4738},[1856],[1797,4740,4742,4762],{"className":4741},[1860,1861],[1797,4743,4745,4759],{"className":4744},[1865],[1797,4746,4748],{"className":4747,"style":1870},[1869],[1797,4749,4750,4753],{"style":1956},[1797,4751],{"className":4752,"style":1878},[1877],[1797,4754,4756],{"className":4755},[1882,1883,1884,1885],[1797,4757,1824],{"className":4758},[1847,1851,1885],[1797,4760,1893],{"className":4761},[1892],[1797,4763,4765],{"className":4764},[1865],[1797,4766,4768],{"className":4767,"style":1900},[1869],[1797,4769],{}," a group effect. Event clustering may require random effects, event-level models, or correlated processes; multiplying independent Poisson probabilities is then invalid.",[1979,4772,4774],{"id":4773},"_7-a-current-data-literacy-example","7. A current data-literacy example",[1793,4776,4777,4778,4781],{},"The Association of British Insurers reported £11.9bn of motor claim payouts across 2.5m claims in 2025. Dividing gives £4,760 per reported claim, but that number should ",[3238,4779,4780],{},"not"," be presented as market severity without checking:",[4137,4783,4784,4787,4790,4793,4796],{},[4140,4785,4786],{},"whether payout and count scopes match;",[4140,4788,4789],{},"claim versus coverage definitions;",[4140,4791,4792],{},"whether figures are paid during year or ultimate accident-year cost;",[4140,4794,4795],{},"mix of damage, theft, injury, and third-party claims;",[4140,4797,4798],{},"the ABI's warning that improved 2025 coverage limits direct year-on-year comparison.",[1793,4800,4801],{},"The example illustrates why a quotient is not automatically a statistical estimand.",[1979,4803,4805],{"id":4804},"_8-validation-checklist","8. Validation checklist",[1984,4807,4808,4818],{},[1987,4809,4810],{},[1990,4811,4812,4815],{},[1993,4813,4814],{},"Check",[1993,4816,4817],{},"What failure suggests",[2010,4819,4820,4828,4836,4844,4852,4860],{},[1990,4821,4822,4825],{},[2015,4823,4824],{},"predicted vs observed counts by exposure band",[2015,4826,4827],{},"offset or non-linearity problem",[1990,4829,4830,4833],{},[2015,4831,4832],{},"Pearson\u002Fdeviance residual pattern",[2015,4834,4835],{},"misspecified mean or variance",[1990,4837,4838,4841],{},[2015,4839,4840],{},"variance-to-mean by homogeneous segment",[2015,4842,4843],{},"overdispersion or omitted grouping",[1990,4845,4846,4849],{},[2015,4847,4848],{},"zero calibration",[2015,4850,4851],{},"excess-zero mechanism",[1990,4853,4854,4857],{},[2015,4855,4856],{},"rolling-time validation",[2015,4858,4859],{},"trend\u002Fseasonality instability",[1990,4861,4862,4865],{},[2015,4863,4864],{},"event-level residual clustering",[2015,4866,4867],{},"dependence",[1979,4869,4871],{"id":4870},"practice","Practice",[4873,4874,4875,4878,4986],"ol",{},[4140,4876,4877],{},"Expected annual rate is 0.08 per policy. What is Poisson expected count for 2,500 policy-years?",[4140,4879,4880,4881,4932,4933,4985],{},"Under ",[1797,4882,4884,4902],{"className":4883},[1800],[1797,4885,4887],{"className":4886},[1804],[1806,4888,4889],{"xmlns":1808},[1810,4890,4891,4899],{},[1813,4892,4893,4895,4897],{},[1819,4894,3736],{},[2072,4896,2173],{},[2861,4898,3778],{},[1826,4900,4901],{"encoding":1828},"\\mu=2",[1797,4903,4905,4923],{"className":4904,"ariaHidden":1834},[1833],[1797,4906,4908,4911,4914,4917,4920],{"className":4907},[1838],[1797,4909],{"className":4910,"style":3432},[1842],[1797,4912,3736],{"className":4913},[1847,1851],[1797,4915],{"className":4916,"style":2112},[2111],[1797,4918,2173],{"className":4919},[2116],[1797,4921],{"className":4922,"style":2112},[2111],[1797,4924,4926,4929],{"className":4925},[1838],[1797,4927],{"className":4928,"style":3268},[1842],[1797,4930,3778],{"className":4931},[1847]," and ",[1797,4934,4936,4955],{"className":4935},[1800],[1797,4937,4939],{"className":4938},[1804],[1806,4940,4941],{"xmlns":1808},[1810,4942,4943,4952],{},[1813,4944,4945,4947,4949],{},[1819,4946,3769],{},[2072,4948,2173],{},[2861,4950,4951],{},"0.5",[1826,4953,4954],{"encoding":1828},"\\alpha=0.5",[1797,4956,4958,4976],{"className":4957,"ariaHidden":1834},[1833],[1797,4959,4961,4964,4967,4970,4973],{"className":4960},[1838],[1797,4962],{"className":4963,"style":3402},[1842],[1797,4965,3769],{"className":4966,"style":4043},[1847,1851],[1797,4968],{"className":4969,"style":2112},[2111],[1797,4971,2173],{"className":4972},[2116],[1797,4974],{"className":4975,"style":2112},[2111],[1797,4977,4979,4982],{"className":4978},[1838],[1797,4980],{"className":4981,"style":3268},[1842],[1797,4983,4951],{"className":4984},[1847],", find negative-binomial variance.",[4140,4987,4988],{},"A portfolio's exposure doubles while rate is unchanged. What happens to expected count and expected rate?",[4990,4991,4993],"legacy-details",{"title":4992},"Answers",[4873,4994,4995,5085,5208],{},[4140,4996,4997,2244],{},[1797,4998,5000,5026],{"className":4999},[1800],[1797,5001,5003],{"className":5002},[1804],[1806,5004,5005],{"xmlns":1808},[1810,5006,5007,5023],{},[1813,5008,5009,5012,5015,5018,5020],{},[2861,5010,5011],{},"2,500",[2072,5013,5014],{},"×",[2861,5016,5017],{},"0.08",[2072,5019,2173],{},[2861,5021,5022],{},"200",[1826,5024,5025],{"encoding":1828},"2{,}500\\times0.08=200",[1797,5027,5029,5058,5076],{"className":5028,"ariaHidden":1834},[1833],[1797,5030,5032,5036,5039,5045,5049,5052,5055],{"className":5031},[1838],[1797,5033],{"className":5034,"style":5035},[1842],"height:0.8389em;vertical-align:-0.1944em;",[1797,5037,3778],{"className":5038},[1847],[1797,5040,5042],{"className":5041},[1847],[1797,5043,2146],{"className":5044},[2446],[1797,5046,5048],{"className":5047},[1847],"500",[1797,5050],{"className":5051,"style":3020},[2111],[1797,5053,5014],{"className":5054},[3024],[1797,5056],{"className":5057,"style":3020},[2111],[1797,5059,5061,5064,5067,5070,5073],{"className":5060},[1838],[1797,5062],{"className":5063,"style":3268},[1842],[1797,5065,5017],{"className":5066},[1847],[1797,5068],{"className":5069,"style":2112},[2111],[1797,5071,2173],{"className":5072},[2116],[1797,5074],{"className":5075,"style":2112},[2111],[1797,5077,5079,5082],{"className":5078},[1838],[1797,5080],{"className":5081,"style":3268},[1842],[1797,5083,5022],{"className":5084},[1847],[4140,5086,5087,2244],{},[1797,5088,5090,5123],{"className":5089},[1800],[1797,5091,5093],{"className":5092},[1804],[1806,5094,5095],{"xmlns":1808},[1810,5096,5097,5120],{},[1813,5098,5099,5101,5103,5105,5107,5113,5115,5117],{},[2861,5100,3778],{},[2072,5102,2854],{},[2861,5104,4951],{},[2072,5106,2085],{"stretchy":2084},[2182,5108,5109,5111],{},[2861,5110,3778],{},[2861,5112,3778],{},[2072,5114,2091],{"stretchy":2084},[2072,5116,2173],{},[2861,5118,5119],{},"4",[1826,5121,5122],{"encoding":1828},"2+0.5(2^2)=4",[1797,5124,5126,5145,5199],{"className":5125,"ariaHidden":1834},[1833],[1797,5127,5129,5133,5136,5139,5142],{"className":5128},[1838],[1797,5130],{"className":5131,"style":5132},[1842],"height:0.7278em;vertical-align:-0.0833em;",[1797,5134,3778],{"className":5135},[1847],[1797,5137],{"className":5138,"style":3020},[2111],[1797,5140,2854],{"className":5141},[3024],[1797,5143],{"className":5144,"style":3020},[2111],[1797,5146,5148,5152,5155,5158,5187,5190,5193,5196],{"className":5147},[1838],[1797,5149],{"className":5150,"style":5151},[1842],"height:1.0641em;vertical-align:-0.25em;",[1797,5153,4951],{"className":5154},[1847],[1797,5156,2085],{"className":5157},[2138],[1797,5159,5161,5164],{"className":5160},[1847],[1797,5162,3778],{"className":5163},[1847],[1797,5165,5167],{"className":5166},[1856],[1797,5168,5170],{"className":5169},[1860],[1797,5171,5173],{"className":5172},[1865],[1797,5174,5176],{"className":5175,"style":3309},[1869],[1797,5177,5178,5181],{"style":2418},[1797,5179],{"className":5180,"style":1878},[1877],[1797,5182,5184],{"className":5183},[1882,1883,1884,1885],[1797,5185,3778],{"className":5186},[1847,1885],[1797,5188,2091],{"className":5189},[2145],[1797,5191],{"className":5192,"style":2112},[2111],[1797,5194,2173],{"className":5195},[2116],[1797,5197],{"className":5198,"style":2112},[2111],[1797,5200,5202,5205],{"className":5201},[1838],[1797,5203],{"className":5204,"style":3268},[1842],[1797,5206,5119],{"className":5207},[1847],[4140,5209,5210],{},"Expected count doubles; expected count per exposure remains unchanged.",[1979,5212,5214],{"id":5213},"source-for-the-current-example","Source for the current example",[4137,5216,5217],{},[4140,5218,5219],{},[5220,5221,5225],"a",{"href":5222,"rel":5223},"https:\u002F\u002Fwww.abi.org.uk\u002Fnews\u002Fnews-articles\u002F2026\u002F2\u002F11.9-billion-paid-out-in-2025-to-support-motorists-across-2.5-million-claims\u002F",[5224],"nofollow","Association of British Insurers — £11.9bn paid to motorists in 2025",{"title":10,"searchDepth":5227,"depth":5227,"links":5228},2,[5229,5230,5234,5235,5238,5239,5240,5241,5242,5243],{"id":1981,"depth":5227,"text":1982},{"id":2052,"depth":5227,"text":2053,"children":5231},[5232],{"id":3364,"depth":5233,"text":3365},3,{"id":3371,"depth":5227,"text":3372},{"id":3698,"depth":5227,"text":3699,"children":5236},[5237],{"id":4131,"depth":5233,"text":4132},{"id":4163,"depth":5227,"text":4164},{"id":4187,"depth":5227,"text":4188},{"id":4773,"depth":5227,"text":4774},{"id":4804,"depth":5227,"text":4805},{"id":4870,"depth":5227,"text":4871},{"id":5213,"depth":5227,"text":5214},"Model claim frequency with exposure, overdispersion, heterogeneity, and excess zeros.","md",{"sidebar":5247},{"order":5233},true,{"title":872,"description":5244},"Z-KqqgcPPrb5Q44ykGk6TTsrvn6q_beODZ8ZWoo8xCQ",[5252,5254],{"title":868,"path":869,"stem":870,"description":5253,"children":-1},"Model large losses with Weibull, Pareto, and threshold exceedances while making tail uncertainty visible.",{"title":876,"path":877,"stem":878,"description":5255,"children":-1},"Fit the likelihood that generated the observations, then validate the decision-relevant centre and tail.",1785754734961]