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

For a random sample of size n from a Normal(μ, σ²) distribution with σ² = 16 known, what is the variance of the sample mean as an estimator of μ, relative to the Cramér-Rao lower bound, when n = 25?

The sample mean has variance 16/25 = 0.64, which equals the Cramér-Rao lower bound σ²/n. It is therefore a fully efficient unbiased estimator of μ.

  1. AVariance 0.64 and exceeds the bound
  2. BVariance 0.64 and equals the boundCorrect
  3. CVariance 0.16 and is below the bound
  4. DVariance 16 and equals the bound
  5. Variance 0.64 and is below the bound

Explanation

Var(X̄) = σ²/n = 16/25 = 0.64. Fisher information for μ is n/σ², so the bound is σ²/n = 0.64. The sample mean attains the bound, so it is efficient.

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