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IAI Actuarial Core Principles · Actuarial Statistics · Random sampling and sampling distributions

Independent observations X1,...,X4 come from a population with mean mu and variance sigma squared. Consider the estimator T = (X1 + 2X2 + 3X3 + 4X4)/10 of mu. Which statement is correct?

T is unbiased with variance 0.3 sigma squared. The weights sum to 10 over 10, so its expectation is mu. The variance is the sum of squared weights, 30, divided by 100. This exceeds the sample mean's 0.25 sigma squared.

  1. AT is biased, with E(T) = 0.4 mu
  2. BT is unbiased and Var(T) = 0.3 sigma squaredCorrect
  3. CT is unbiased and Var(T) = 0.25 sigma squared
  4. DT is unbiased and Var(T) = 3 sigma squared
  5. T is biased, with E(T) = 10 mu

Explanation

E(T)=(1+2+3+4)mu/10=mu, so T is unbiased. Var(T)=(1+4+9+16)sigma squared/100=0.3 sigma squared. The sample mean has smaller variance 0.25 sigma squared, so option with 0.25 describes X-bar, not T.

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