Question: 31. An estimator is said to be consistent if for any ! 0, P( !) 0 0 as n

31. An estimator ˆ

is said to be consistent if for any ! 0, P(⏐ˆ

⏐ !) 0 0 as n 0 . That is, ˆ

is consistent if, as the sample size gets larger, it is less and less likely that ˆ

will be further than ! from the true value of . Show that X is a consistent estimator of when 2 by using Chebyshev’s inequality from Exercise 44 of Chapter 3. [Hint:

The inequality can be rewritten in the form P(⏐Y Y⏐ !) 2 Y /!

Now identify Y with X.]

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