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

Let X1,...,Xn be a random sample from an exponential distribution with mean μ. The estimator μ̂ = X̄ is unbiased. Using the Cramér–Rao lower bound for unbiased estimators of μ, which statement is correct?

The Cramér–Rao bound is μ²/n and X̄ attains it. The Fisher information for the sample is n/μ², and the variance of the sample mean of exponentials is μ²/n, so X̄ is a minimum variance unbiased estimator.

  1. AThe bound is μ²/n² and X̄ does not attain it
  2. BThe bound is μ²/n and X̄ attains itCorrect
  3. CThe bound is μ/n and X̄ attains it
  4. DThe bound is μ²/n and X̄ does not attain it
  5. The bound is 1/(nμ²) and X̄ attains it

Explanation

log f = −log μ − x/μ. Second derivative wrt μ is 1/μ² − 2x/μ³; its expectation is 1/μ² − 2/μ² = −1/μ², so I(μ) = n/μ² and the bound is μ²/n. Var(X̄) = μ²/n, which equals the bound, so X̄ attains it.

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