Let X1, X2, ...., Xn be uniformly distributed on the interval (0 to a). Recall that the maximum likelihood estimator of a is â =max(Xi).
a) Argue intuitively why â cannot be an unbiased estimator for a.
b) Suppose that E(â ) = na /(n+1).
Is it reasonable that â 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 Y≤ y if and only if each Xi ≤ y to derive the cumulative distribution function of Y. Then show that the probability density function of Y is: f(y) = nyn-1/ an for (0 ≤ y ≤ a) and f(y) = 0 otherwise. 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 â1 = 2Χbar and
â2 = [(n+1)/n] max (Xi), where max(Xi) is the largest observation in a random
sample of size n. It can be shown that V(â1) = a2/3n and that V(â2) = a2/[n(n+2)].
Show that if n>1, â2 is a better estimator than â1. In what sense is it a better estimator of a?
The solution determines the point estimator for a uniformly distributed interval.