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IAI Actuarial Core Principles · Actuarial Statistics · Bayesian inference: priors, posteriors, loss functions and credible intervals

The posterior distribution of a parameter θ is Exponential with rate 0.5 per unit (so the density is 0.5e^(-0.5θ) for θ>0). Under absolute error loss, the Bayes estimate of θ is closest to which value?

Absolute error loss leads to the posterior median. For an Exponential distribution with rate 0.5 the median is ln2 divided by 0.5, which is about 1.39.

  1. A1.39Correct
  2. B2.00
  3. C0.69
  4. D4.00
  5. 1.00

Explanation

Absolute error loss gives the posterior median m. Solve 1 - e^(-0.5m) = 0.5, so m = ln2/0.5 = 1.386. The mean 2 is the answer under quadratic loss. 0.69 forgets to divide by the rate.

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