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IAI Actuarial Core Principles · Actuarial Statistics · Credibility theory: Bayesian and empirical Bayes credibility premiums

Empirical Bühlmann estimation uses data on 3 risks, each with 4 years of claims. Risk means are 10, 14 and 12. Within-risk sample variances (divisor 3) are 6, 8 and 10. Estimating E[s²(θ)] by the average of the sample variances and Var[m(θ)] by (variance of risk means, divisor 2) − E[s²]/n, what is the estimated credibility factor Z?

Z is 0.50. The average within-risk variance is 8, and the variance of risk means is 4. Subtracting 8/4 = 2 gives Var[m] = 2, so k = 4 and Z = 4/8 = 0.5.

  1. A0.25
  2. B0.40
  3. C0.50
  4. D0.67Correct
  5. 0.80

Explanation

E[s²] = (6+8+10)/3 = 8. Mean of risk means = 12; variance with divisor 2 = (4+4+0)/2... check: deviations -2, 2, 0 give squares 4,4,0, sum 8, divided by 2 = 4. Var[m] = 4 − 8/4 = 2. k = 8/2 = 4, Z = 4/(4+4) = 0.5. So the correct value is 0.50, not 0.67, which would come from forgetting to subtract E[s²]/n (k = 8/4 = 2).

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