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

A random sample of size n is taken from a Normal(μ, σ²) distribution. Consider the estimator S*² = (1/n)Σ(Xi - X̄)² of σ². Which statement is correct?

S*² is biased for finite n, with bias -σ²/n, but both bias and variance (2(n-1)σ⁴/n²) tend to zero. So the mean square error tends to zero and the estimator is consistent. Bias for finite samples does not prevent consistency.

  1. AS*² is biased for every n and is not consistent
  2. BS*² is unbiased and consistent
  3. CS*² is unbiased but not consistent
  4. DS*² is biased for finite n, but its bias and variance both tend to 0, so it is consistentCorrect
  5. S*² is biased, and consistent only if μ is known

Explanation

E[S*²] = (n-1)σ²/n, so bias = -σ²/n which tends to 0. Var(S*²) = 2(n-1)σ⁴/n², which tends to 0. Therefore MSE tends to 0 and the estimator is consistent, even though it is biased for each finite n. Knowing μ is not required.

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