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

For a random sample of size n from a population with mean μ and variance σ², consider the estimator of μ given by T = (X1 + ... + Xn)/(n+1). Which expression gives its mean squared error?

The MSE is (nσ² + μ²)/(n+1)². The variance is nσ²/(n+1)² and the bias is -μ/(n+1), whose square is μ²/(n+1)². Adding these gives the stated expression.

  1. A(nσ² + μ²)/(n+1)²Correct
  2. Bσ²/n
  3. C(σ² + μ²)/(n+1)²
  4. Dnσ²/(n+1)² + μ²/(n+1)
  5. nσ²/(n+1)

Explanation

E[T] = nμ/(n+1), so bias = -μ/(n+1) and bias² = μ²/(n+1)². Var(T) = nσ²/(n+1)². MSE = (nσ² + μ²)/(n+1)². Option σ²/n ignores both the shrinkage and the bias.

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