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FRM Part I · FRM Exam Part I · Sample Moments

Returns are independent with common mean μ, but X1 has variance 4 and X2 has variance 12. An analyst forms the unbiased linear estimator wX1 + (1−w)X2. What weight w gives the minimum-variance estimator, and what is that minimum variance?

The minimum-variance weight is 0.75 on X1, proportional to inverse variance, giving variance 4(0.5625)+12(0.0625)=3.00. Equal weights give 4.00, so with heteroskedastic observations the simple sample mean is not the best linear unbiased estimator.

  1. Aw = 0.50, variance 4.00
  2. Bw = 0.75, variance 3.00Correct
  3. Cw = 0.25, variance 3.00
  4. Dw = 0.75, variance 4.00

Explanation

Var = 4w² + 12(1−w)². Setting the derivative to zero: 8w − 24(1−w) = 0 gives w = 0.75. Variance = 4(0.5625) + 12(0.0625) = 2.25 + 0.75 = 3.00. Equal weighting gives 2 + 3 = ... 1+3 = 4, which is higher, so the sample mean is not BLUE with unequal variances.

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