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

X1, X2, X3 are independent observations from a distribution with mean μ and variance σ². Which of the following estimators of μ is biased?

The estimator (X1+X2+X3)/4 is biased. Its expectation is 3μ/4, not μ, because the weights sum to 3/4 rather than 1. All the other estimators are linear combinations with weights summing to one, so they are unbiased for μ.

  1. A(X1 + X2 + X3)/3
  2. B(X1 + 2X2 + 3X3)/6
  3. C(2X1 + X2)/3
  4. D(X1 + X2 + X3)/4Correct
  5. X2

Explanation

An estimator a1X1+a2X2+a3X3 is unbiased for μ iff the weights sum to 1. Weights sum: option A 1, B (1+2+3)/6=1, C 1, E 1. Option D has weights summing to 3/4, so its expectation is 0.75μ, giving bias -0.25μ.

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