Exam MAS-I Practice Question 36

Study guide: Exam MAS-I

A random sample of $n = 10$ claim severities is drawn from a normal distribution with unknown mean and variance $\sigma^2$ . Two estimators of $\sigma^2$ are under consideration:

the sample variance $S^2 = \frac{1}{9}\sum (X_i - \bar{X})^2$ , which is unbiased with $\text{Var}(S^2) = \frac{2\sigma^4}{9}$ ;
the maximum likelihood estimator $\hat\sigma^2 = \frac{1}{10}\sum (X_i - \bar{X})^2 = 0.9\,S^2$ .

Part a (1 point)

Calculate the mean square error of $\hat\sigma^2$ , as a multiple of $\sigma^4$ . Give the multiple to two decimal places (for example, enter 0.25 if the MSE is 0.25 times $\sigma^4$ ).

Part b (1 point)

Using mean square error as the criterion, which statement is correct?

A) $S^2$ is preferred, because no biased estimator can have a smaller MSE than an unbiased one
B) $\hat\sigma^2$ is preferred, because its MSE is below that of $S^2$
C) $S^2$ is preferred, because a relative efficiency $\text{Var}(S^2)/\text{Var}(\hat\sigma^2)$ above 1 favours it
D) Neither is preferred, because one is a constant multiple of the other
E) $S^2$ is preferred, because the bias of $\hat\sigma^2$ grows as $n$ increases

Part c (1 point)

A colleague proposes to rank the two estimators by their relative efficiency, $\text{Var}(S^2)/\text{Var}(\hat\sigma^2)$ . Critique this proposal, and describe one situation in which $S^2$ would still be the better choice.

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Exam MAS-I Practice Question 36

Part a (1 point)

Part b (1 point)

Part c (1 point)

Concepts tested

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