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IAI Actuarial Core Principles · Actuarial Statistics · Estimators and their properties

Claim amounts are exponential with mean θ. From a sample of n claims, the estimator θ̂ = X̄ is used. For what smallest sample size n is the variance of X̄ at most 0.01θ²?

The variance of the sample mean is θ²/n because an exponential with mean θ has variance θ². Setting θ²/n ≤ 0.01θ² gives n ≥ 100. The smallest sample size is therefore 100.

  1. A10
  2. B50
  3. C100Correct
  4. D1,000
  5. 25

Explanation

For an exponential distribution with mean θ the variance is θ², so Var(X̄) = θ²/n. Requiring θ²/n ≤ 0.01θ² gives n ≥ 100. Since X̄ is unbiased, the MSE equals this variance and tends to 0 as n grows, showing consistency. The choice 10 would give 0.1θ².

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