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

A random sample X1, X2, X3 is drawn from a distribution with mean μ and variance σ². Consider the estimator T = (X1 + 2X2 + 3X3)/5 for μ. Which statement is correct?

T is biased with bias μ/5 and variance 14σ²/25. The weights sum to 6, so the expected value is 6μ/5, not μ. The variance is the sum of squared weights, 14, times σ² divided by 25.

  1. AT is unbiased and has variance 14σ²/25
  2. BT is biased with bias μ/5 and variance 14σ²/25Correct
  3. CT is unbiased and has variance 6σ²/25
  4. DT is unbiased and has variance 14σ²/5
  5. T is biased with bias μ/5 and variance 6σ²/25

Explanation

E[T] = (1+2+3)μ/5 = 6μ/5, so the bias is μ/5 and T is biased. Var(T) = (1+4+9)σ²/25 = 14σ²/25. Options claiming unbiasedness wrongly divide the weights' sum by 5 as if it were 6.

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