IAI Actuarial Core Principles · Actuarial Statistics · Bayesian inference: priors, posteriors, loss functions and credible intervals
The posterior distribution of a claim rate parameter λ is Gamma with shape 12 and rate 4 (mean 3, variance 0.75). Under quadratic (squared error) loss, what is the Bayes estimate of λ?
The Bayes estimate under squared error loss is the posterior mean. For a Gamma distribution with shape 12 and rate 4 the mean is 12 divided by 4, which equals 3.
- A2.75
- B3.00Correct
- C3.25
- D0.75
- 12.00
Explanation
Under squared error loss the Bayes estimate is the posterior mean. For Gamma(shape 12, rate 4) the mean is 12/4 = 3. The value 0.75 is the posterior variance, which is not the estimate, and 2.75 is the mode (11/4), which is the answer under all-or-nothing loss.
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