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FRM Part I · FRM Exam Part I · Sample Moments

Observations X1, X2, X3, X4 are i.i.d. with mean mu and variance 20. An analyst uses the estimator (X1 + X2 + X3 + X4)/4 for mu. What is the variance of this estimator, and how does it compare with the variance of the estimator that uses only X1?

The variance of the sample mean of four i.i.d. observations is 20 divided by 4, which equals 5. The estimator using only the first observation is also unbiased but has variance 20, so the sample mean has one quarter of that variance.

  1. AVariance 5, which is one quarter of the single-observation varianceCorrect
  2. BVariance 10, which is one half of the single-observation variance
  3. CVariance 5, which is one half of the single-observation variance
  4. DVariance 80, which is four times the single-observation variance

Explanation

For i.i.d. data, Var of the sample mean = sigma^2/n = 20/4 = 5. The estimator using X1 alone has variance 20 and is also linear and unbiased. The sample mean's variance is a quarter of that, consistent with it being the minimum-variance linear unbiased estimator.

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