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

A sample of size n is taken from a uniform distribution on (0, θ). Estimator A = 2X̄ is unbiased with variance θ²/(3n). Estimator B = ((n+1)/n)·max(Xi) is unbiased with variance θ²/(n(n+2)). For n = 4, what is the ratio MSE(A)/MSE(B)?

The ratio is 2.00. Both estimators are unbiased, so MSE equals variance. For n=4, A has variance θ²/12 and B has θ²/24, so dividing gives 2, showing B is twice as efficient.

  1. A2.00Correct
  2. B1.00
  3. C3.00
  4. D0.50
  5. 4.00

Explanation

For n=4: Var(A) = θ²/12. Var(B) = θ²/(4·6) = θ²/24. Both are unbiased, so MSE equals variance. Ratio = (1/12)/(1/24) = 2.

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