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

Which statement about the estimator S² = (1/n)Σ(Xi − X̄)² of σ², for a random sample of size n from a normal distribution, is correct?

The estimator has bias −σ²/n and is consistent. Its expectation is (n−1)σ²/n, which is below σ², but both bias and variance vanish as n increases, so it converges in probability to σ².

  1. AIt is unbiased and consistent
  2. BIt is biased with bias −σ²/n and is consistentCorrect
  3. CIt is biased with bias σ²/n and is inconsistent
  4. DIt is unbiased but inconsistent
  5. It is biased with bias −σ²/n and is inconsistent

Explanation

E[S²] = (n−1)σ²/n, so bias = −σ²/n. The bias tends to 0 and the variance tends to 0 as n grows, so the estimator is consistent (asymptotically unbiased). The inconsistent options ignore this.

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