# BA3202-L2 - BA3202 Actuarial Statistics Lecture 1 Summary...

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BA3203 L2 BA3202 Actuarial Statistics Lecture 1 Summary: - Decision Theory - Bayesian Statistics

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BA3203 L2 Decision Theory Game Theory Domination Minimax Criterion & saddle point Optimal randomised strategy Statistical games Strategy functions Construct expected loss matrix Bayes criterion Actuarial Science NTU Jade Nie 2
BA3203 L2 Bayesian statistics Prior/posterior distribution Posterior pdf of 𝜽 : 𝒇 𝜽 𝑿 = 𝒇 𝑿 𝜽 𝒇 𝜽 𝒇 𝑿 ∝ 𝒇 𝑿 𝜽 𝒇 𝜽 ? ? 𝜃 = ?(? ? |𝜃) 𝑛 ?=1 : the likelihood of the sample is a function of unknown 𝜃 ? ? = ∫ ? ? 𝜃 ? 𝜃 ?𝜃 : a constant that does not involve 𝜃 Common loss functions Quadratic loss: ? ? ? , 𝜃 = ? ? − 𝜃 2 Bayesian estimator: ? = ?(𝜃|? ) , mean of posterior Absolute error loss: ? ? ? , 𝜃 = ? ? − 𝜃 Bayesian estimator: median of posterior “All or nothing” error: ? ? ? , 𝜃 = 0, ??? ? = 𝜃 1, ??? ? ≠ 𝜃 Bayesian estimator: mode of posterior Actuarial Science NTU Jade Nie 3

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BA3203 L2 BA3202 Actuarial Statistics Lecture 2: - Loss distributions
BA3203 L2 Objectives 1. Describe the properties of the statistical distributions which are suitable for modelling individual and aggregate losses. 2. Derive moments and moment generating functions (where defined) of loss distributions including the gamma, exponential, Pareto, generalised Pareto, normal, lognormal, Weibull and Burr distributions. 3. Apply the principles of statistical inference to select suitable loss distributions for sets of claims. Actuarial Science NTU Jade Nie 5

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BA3203 L2 Insurance overview Insurance losses: Frequency of loss: how often does a loss occur over the insured period Severity of loss: how much does each loss cost? Expected total loss=expected no. of loss * expected average severity General procedure for modelling of loss frequency and severity: There is a large sample of candidate “loss distribution” models Each loss distribution is fit on historical losses for the insurance company. All distributions fitted on the data, are compared using goodness‐of‐fit (GOF) criteria. The loss distribution with the best fitness is chosen. The procedure above is done separately for the frequency and the severity of losses. Actuarial Science NTU Jade Nie 6
BA3203 L2 Statistical overview We make the assumptions that historical losses (claims) come from a familiar distribution.

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