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

X1,...,Xn are independent N(μ, σ²) with μ known to be 0. The estimator S = (1/n)ΣXi² is used for σ². Another analyst uses the estimator V = (1/(n+1))ΣXi². Which statement is correct?

S is unbiased and V has bias -σ²/(n+1). With a known zero mean each Xi² has expectation σ², so S is unbiased. V divides the same sum by n+1, giving expectation nσ²/(n+1), which is below σ² by σ²/(n+1).

  1. AS is biased and V is unbiased
  2. BBoth are unbiased
  3. CS is unbiased and V has bias -σ²/(n+1)Correct
  4. DS is unbiased and V has bias +σ²/(n+1)
  5. Both are biased, with S having bias -σ²/n

Explanation

E[Xi²] = σ² since μ = 0, so E[S] = σ² and S is unbiased. E[V] = nσ²/(n+1), so bias = nσ²/(n+1) − σ² = −σ²/(n+1). V underestimates; the sign is negative.

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