Quiz #4 STAT 535

Quiz #4

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answer
B

Considering the situation of observing iid random variables from a symmetric distribution, which of the following statements is the best explanation of why the sample median can be better than the sample mean as an estimator for the mean of the distribution.

[ A ] While the sample mean and the sample median are both unbiased estimators of the mean, the sample median can have a larger variance than the sample mean if the distribution is a heavy-tailed distribution.
[ B ] While the sample mean and the sample median are both unbiased estimators of the mean, the sample median can have a smaller variance than the sample mean if the distribution is a heavy-tailed distribution.
[ C ] The sample median is less resistant to extreme values than is the sample mean.
[ D ] Even though the sample mean is more sensitive to extreme values than is the sample median, the law of large numbers makes it a better estimator (and so the sample median cannot be a better estimator of the distribution mean).

While [ A ] is a true statement (if the distribution is symmetric and not extremely heavy-tailed), it's not an explanation of why the sample median can be better than the sample mean --- one should not prefer the sample median if its variance is larger. [ B ] is correct --- the sample median can have a smaller variance in some situations with a heavy-tailed distribution, and one should prefer the unbiased estimator with the smaller variance. [ C ] is a false statement --- the sample median is much more resistant to extreme values than is the sample mean. [ D ] is also a false statement. While the law of large numbers gives us that the sample mean converges to the correct value, with a symmetric distribution the sample median will also converge to the correct value --- the law of large numbers does not give us that the sample mean is necessarily superior, and in some cases the sample median can be much better. (See the sentence just prior to the footnote on p. 176 of S&W.)