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

Given θ, claim sizes are Exponential with mean 1/θ. The prior for θ is Gamma(3, 600). Four claims total 1,000 rupees (in thousands, use the same units as 600). Under the absolute error loss, which statement about the Bayes estimate is correct, noting the posterior is Gamma(7, 1600)?

Under absolute error loss the Bayes estimate is the posterior median. For the right-skewed Gamma(7, 1600) posterior the median lies below the mean of 0.004375, so the statement about the median being below the mean is correct.

  1. AIt is the posterior mean 7/1600 = 0.004375
  2. BIt is the posterior mode 6/1600 = 0.00375
  3. CIt is the posterior median, which lies below the meanCorrect
  4. DIt is the posterior median, which lies above the mean
  5. It is the prior mean 3/600 = 0.005

Explanation

Absolute error loss gives the posterior median. A Gamma distribution is right-skewed, so its median is below the mean (0.004375) and above the mode (0.00375). The mean belongs to quadratic loss and the mode to all-or-nothing loss.

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