Consider a random sample $x_1,x_2,$dots,$x_n$ of size $n$ from the Uniform $(0,)$ distri$-$

bution, where $>0.$

$($i$)$ Show that the p$.$d$.$f$.q(t$;$)$ of the sufficient statistic

$T(x)-=ma{x}_{1}in{x}_i$

is given by

$q(t$;$)=\frac{n{t}^{n-1}}{{}^n},0t$

Deduce expressions for the mean and variance of $T.$

$($ii$)$ Show that the method of moments gives the estimator ${}^{**}-=2\stackrel{}{x}$ of $.$ Comment

on this estimator with regard to the sufficiency principle and show that it can

yield estimates of $$ that are incompatible with the sample data, i$.$e$.$ that there

are cases in which it is not possible for ${}^{**}$ to attain the value $.$

$($iii$)$ Write down the maximum likelihood estimator of $$ and show that it is not

unbiased.

$($iv$)$ Find the constant a such that $aT$ is an unbiased estimator of $.$

$($v$)$ Find the mean squared errors of the estimators of parts $($ii$),($iii$)$ and $($iv$),$

respectively, and comment on their relative values.