Q$-1(4+5+6+7+8+10+10)($a$)$ Let $x_{i}N({}_i,1)$ with ${}_i={}_i,i=1,$dots,$n,$ estimated

as widehat$({)}_i={}_i$widehat$(),$ where ${}_i$ are known constants and $$ is the parameter with its estimator

widehat$()={\displaystyle \underset{i=1}{\overset{n}{}}}{}_i\frac{x_i}{{\displaystyle \underset{i=1}{\overset{n}{}}}}{}_i^2$

Show that widehat$()$ and ${\displaystyle \underset{i=1}{\overset{n}{}}}(x_i-$widehat$({)}_i{)}^2$ are independent.

$($b$)$ Let $x_{i}N(,{}^2),i=1,2,$ iid. Find the amount of information about $$ and ${}^2$ in

$x_1+x_2$ and $x_1-x_2.$

$($c$)$ Let $x_{i}Bernoulli(p),i=1,$dots,$n,$ iid. Consider two estimators of $p$ as

widehat$(p{)}_1=\frac{S}{n},$widehat$(p{)}_2=\frac{S+1}{n+2}$

where $S={\displaystyle \underset{i=1}{\overset{n}{}}}{x}_i.$ Compare widehat$(p{)}_1$ and widehat$(p{)}_2$ using an appropriate risk function.

$($d$)$ Let $x_{i}N({}_1,{}^2),i=1,$dots,$m,$ iid, and $Y_{j}N({}_2,2{}^2),j=1,$dots,$n,$ iid, be

two independent random samples with all parameters unknown. Find a complete sufficient

statistic of $({}_1,{}_2,{}^2{)}^{\text{'}}.$ Compute UMVUEs of $({}_1-{}_2{)}^2$ and of ${}^2.$

$($e$)$ Let $x_1,$dots,$x_{n}Exp()$ so that $f(x$; Consider two

competing estimators of $,$widehat$({)}_1=-logwidehat()$ and widehat$({)}_2=\frac{1}{\stackrel{}{x}},$ where

widehat$()=\frac{1}{n}$#$\{in$:$x_{i}1\},\stackrel{}{x}=\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{}}}{x}_i.$

Determine the ARE of widehat$({)}_1$ vs$.$ widehat$({)}_2.[$If needed, you may use $E(x)=\frac{1}{},$Var$(x)=\frac{1}{{}^2}.]$

$($f$)$ In $($d$),$ let ${}_1={}_2=.$ Find MLE, widehat$(),$ of the vector $=(,{}^2{)}^{\text{'}}.$ Compute Fisher

information matrix and variance$-$covariance matrix of widehat$().$

$($g$)$ Let $Y_{i}N({}_i,{}^2),i=1,$dots,$n,$ with ${}_i={}_0+{}_1{x}_i,{}_0,{}_1$inR and $x_i$ fixed. Assume ${}_0$

and ${}^2$ known constants $($you can take e$.$g$.{}_0=0,{}^2=1$ if you like$).$ Taking ${}_{1}N(0,{}^2)$

as prior, computer the posterior distribution of ${}_1$ in as simplified form as possible.