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Jan 2021 Chapter 4 Empirical Bayes Parameter Estimation 1 UNIVERSITI TUNKU ABDUL RAHMAN Department of Mathematics and Actuarial Science CONTENTS 4 Empirical Bayes Parameter Esti- mation 2 4.1 Introduction . . . . . . . . . . . . 2 4.2 Nonparametric Estimation . . . . . 3 4.2.1 B¨uhlmann Straub Model: . 3 4.2.2 Credibility-Weighted-Average 13 4.2.3 B¨uhlmann model-Estimating μ , v , and a . . . . . . . . . 16 4.2.4 Data with only One Poli- cyholder: . . . . . . . . . . 25 4.3 Empirical Bayes Semi-Parametric . 28 4.3.1 Poisson Model . . . . . . . 28 4.3.2 Geometric Model . . . . . . 39 UECM3473 Credibility Theory Jan 2021 Chapter 4 Empirical Bayes Parameter Estimation 2 4 Empirical Bayes Parameter Estima- tion 4.1 Introduction In the previous Chapters, the parameters ( μ, v, a ) needed to determine the credibility weighted pre- mium are all given. In this chapter, we will study two methods to estimate these parameters. Empirical Credibility Nonparametric estimation: This is based on unbiased estimators of μ , v and a , Semiparametric estimation: f ( x | θ ) is para- metric (usually Poisson or Geometric) but the π ( θ ) is nonparametric. UECM3473 Credibility Theory
Jan 2021 Chapter 4 Empirical Bayes Parameter Estimation 3 4.2 Nonparametric Estimation 4.2.1 B ¨ uhlmann Straub Model: For each policyholder i, 1 i r (and r > 1), we have observations X i = ( X i 1 , X i 2 , . . . , X i,n i ) of loss per exposure unit corresponding to expo- sures m i = ( m i 1 , m i 2 , . . . , m i,n i ) and n i > 1. This means that m ij X ij is the aggregate loss for period (or unit) j from policyholder i . Let m i = n i j =1 m ij be the total exposure for poli- cyholder i . Be careful to distinguish between X ij (a rate) and m ij X ij (an amount). UECM3473 Credibility Theory Jan 2021 Chapter 4 Empirical Bayes Parameter Estimation 4 Estimation of B ¨ uhlmannStraub param- eters μ, v , and a : STEP 1. Calculate the sample mean ¯ x i and biased σ 2 i ( σ 2 in TI-30). ¯ x i = 1 m i n i summationdisplay j =1 m ij x ij ; σ 2 i = n i summationdisplay j =1 m ij ( x ij - ¯ x i ) 2 m i then the (unbiased) sample variance for each policyholder. ˆ v i = 1 n i - 1 n i summationdisplay j =1 m ij ( x ij - ¯ x i ) 2 = m i n i - 1 σ 2 i UECM3473 Credibility Theory
Jan 2021 Chapter 4 Empirical Bayes Parameter Estimation 5 STEP 2. Calculate the weighted average of the sample variances ˆ v i with weights n i - 1. If the poli- cyholders all have the same number of periods of exposure, this is exactly the average of the sample variances. ˆ v = r i =1 ( n i - 1)ˆ v i r i =1 ( n i - 1) STEP 3. Calculate ˆ μ = ¯ x = r i =1 m i ¯ x i m and biased σ 2 = r i =1 m i x i - ¯ x ) 2 m then, UECM3473 Credibility Theory Jan 2021 Chapter 4 Empirical Bayes Parameter Estimation 6 ˆ a = r i =1 m i ( x i - ¯ x ) 2 - ( r - 1)ˆ v m - m - 1 r i =1 m 2 i = 2 - ( r - 1)ˆ v m - m - 1 r i =1 m 2 i The estimator ˆ a may be negative. If it is neg- ative but small in absolute value, the authors suggest setting it to zero and Z i = 0 for all i . The estimator for all risks is ˆ μ = ¯ x in this case. After estimating the B¨uhlmann parameters, we estimate a given client’s credibility pre- mium based on its own experience as ˆ Z i ¯ x i + (1 - ˆ Z i μ where ˆ k = ˆ v ˆ a and ˆ Z i = m i m i + ˆ k = m i ˆ a m i ˆ a + ˆ v UECM3473 Credibility Theory
Jan 2021 Chapter 4 Empirical Bayes Parameter Estimation 7 Example 1.

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