Question: Let X1, X2, , Xn be uniformly distributed on the interval 0 to a. Recall that the maximum likelihood estimator of a is . (a)

Let X1, X2, , Xn be uniformly distributed on the interval 0 to

a. Recall that the maximum likelihood estimator of a is .

(a) Argue intuitively why cannot be an unbiased estimator for a.

(b) Suppose that . Is it reasonable that consistently underestimates a? Show that the bias in the estimator approaches zero as n gets large.

(c) Propose an unbiased estimator for a.

(d) Let Y max(Xi). Use the fact that if and only if each to derive the cumulative distribution function of Y. Then show that the probability density function of Y is Use this result to show that the maximum likelihood estimator for a is biased.

(e) We have two unbiased estimators for a: the moment estimator and , where max(Xi) is the largest observation in a random sample of size n. It can be shown that and that . Show that if n 1, is a better estimator than . In what sense is it a better estimator of a?

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