# MATH4427 L4 ver4 Part1.pdf - MATH4427 Loss Models and their...

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1MATH4427 Loss Model and their applicationsLecture Note 4: Parameter estimation in parametric modelMATH4427 Loss Models and their applicationsLecture Note 4Parameter Estimation in parametric model
2MATH4427 Loss Model and their applicationsLecture Note 4: Parameter estimation in parametric modelIntroductionRecall that the construction of a loss model consists of three main stages:Stage 1:Choose probabilitydistributions for claimseverity?and claimfrequency?Stage 2:Estimate the parametervalues in the probabilitydistributions of?and?.(Parameter estimation)Stage 3:Study the aggregateloss𝑆or relatedquantities using suitablemodel.ParetoGammaLognormalClaimseverity?Tail distribution+ nature of riskPoissonBinomialNegative binomialClaimfrequency??????−1?~????? ሺ?, 𝜃ሻ?~𝑃??????ሺ?ሻEstimate the values of?, 𝜃and?using thesample data𝑆 = ?1+ ?2+ ⋯ + ??(where???areindependent)𝑆 = ?1+ ?2+ ⋯ + ?𝑁(where??s are iid)Individual risk modelCollective risk model
3MATH4427 Loss Model and their applicationsLecture Note 4: Parameter estimation in parametric modelSo far, we have explored how to execute the Stage 1 (model selection) andStage 3 (Aggregate loss model) in Chapter 2 and Chapter 3 respectively. Inorder to complete the construction of the loss model, we will study how toestimate the parameter values in the probability distributions (tuning) usingthe sample data and the following methods:Case 1: The probability distribution is fixed (parameter is fixed) butunknownMethod of momentsMaximum likelihood methodCase 2: The probability distribution isnot fixed(parameters are randomvariables)Bayesian estimation: How to update the probability distribution of theparameter𝜃when there is an updated sample data (prior distribution andposterior distribution)? How to estimate the value of the parameter fromthe distribution (calledBayes estimate)?
4MATH4427 Loss Model and their applicationsLecture Note 4: Parameter estimation in parametric modelHow to do the estimation? Estimation through random samplingIn most real-life statistical problems, the size of population is large for us toinvestigate them all. Instead, we try to draw conclusions about the populationby examining a portion of the population (calledrandom sample).Estimated Mean= ?ƸEstimated Standard Deviation= 𝜎???? = ?(Unknown)Standard Deviation= 𝜎(Unknown)PopulationSampleRandomSamplingInvestigationPrediction
5MATH4427 Loss Model and their applicationsLecture Note 4: Parameter estimation in parametric modelThis process is calledStatistical Interference. It is the process by whichinformation from sample data is used to draw conclusions about thepopulation from which the sample was selected.Since the sample israndomlydrawn from the whole population, theestimations obtained from the sample data are alsorandom quantitiesingeneral. Thus the accuracy (or reliability) of the estimation method becomes animportant issue.

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Term
Winter
Professor
NoProfessor
Tags
Maximum likelihood, Estimation theory