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FRM Part I · FRM Exam Part I · Sample Moments

Two independent samples estimate the same population mean. Sample A has 40 observations and sample B has 160 observations, both from i.i.d. draws with the same variance. A combined estimator uses weights chosen to minimize the variance of the pooled estimate of the mean. Which statement is correct about the minimum-variance combination of the two sample means, and the ratio of its standard error to that of sample A's mean?

Weight sample A at 0.2 and sample B at 0.8, which equals pooling all 200 observations. The variance falls to one-fifth of A's variance, so the standard error ratio is the square root of 0.2, about 0.447. Reporting 0.200 confuses variance with standard error.

  1. AWeight A at 0.5 and B at 0.5; ratio of standard error is 0.707
  2. BWeight A at 0.2 and B at 0.8; ratio of standard error is 0.447Correct
  3. CWeight A at 0.8 and B at 0.2; ratio of standard error is 0.894
  4. DWeight A at 0.2 and B at 0.8; ratio of standard error is 0.200

Explanation

Minimum variance weights are proportional to n: 40/200 = 0.2 and 160/200 = 0.8; this equals the pooled mean of 200 observations. Variance is sigma^2/200 versus sigma^2/40 for A, so the ratio of variances is 0.2 and the ratio of standard errors is sqrt(0.2) = 0.447. The 0.200 option reports the variance ratio, not the standard error ratio.

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