Chapter 7.4, Problem 52E

a.

To determine

Explain the reason of $\widehat{a}$ not being an unbiased estimator for a.

b.

To determine

Check whether it is reasonable that $\widehat{a}$ consistently underestimated a, or not.

Show that the bias in the estimator approaches zero as n gets large.

c.

To determine

Propose an unbiased estimator for a.

d.

To determine

Derive the cumulative distribution function of Y.

Show that the probability density function of Y is $f\left(y\right)=\{\begin{array}{l}\frac{n{y}^{n-1}}{a^n},0\le y\le a\\ 0,\text{otherwise}\end{array}$

Show that the maximum likelihood estimator of a is biased.

e.

To determine

Show that if $n>1$, ${\widehat{a}}_2$ is a better estimator than $\widehat{a}$.

Explain in what sense it is a better estimator of a.