Q$-2(7+10+7+10+10+4+2)$ Let $x_1,x_2,$dots,$x_n$ be a random sample from a distribution

with probability density function

$f(x$;$,v)=\frac{v^{}}{x^{+1}},xv,>2,v>0.$

It is known that

$E(x)=\frac{v}{+1},$Var$(x)=\frac{v^2}{(-1{)}^2(-2)}.$

First, suppose that $$ is unknown, but $v$ is known.

$($a$)$ Find the UMP level $$ test for testing $H_0$:$={}_2$ versus $H_1$:$={}_1,$ where

$($b$)$ Find the UMP level $$ test for testing $H_0$:${}_1$ versus $H_1$:${}_1,$ where ${}_1>2.$

$($c$)$ Without deriving the expression of MLE of $,$ find an asymptotic size $$ likelihood ratio

test for testing $H_0$:$={}_1$ versus $H_1$:${}_1,$ where ${}_1>2.($For simplicity, you do not

need to verify the second$-$order condition.$)$

$($d$)$ Find an asymptotic size $$ score test for testing $H_0$:$={}_1$ versus $H_1$:${}_1.$

$($e$)$ Construct an asymptotic two$-$sided $1-$ Wald interval for $.$

$($f$)$ It is known that $Y={\displaystyle \underset{i=1}{\overset{n}{}}}log(\frac{x_i}{v})$ follows an Erlang distribution with density

$f(y)=\frac{y^{n-1}\text{e}\text{x}\text{p}(-y)}{(n-1)!}.$

Find an exact two$-$sided $1-$ confidence interval for $.$

$($g$)$ Now suppose that $v$ is unknown but $$ is known, and you want to test $H_0$:$v=v_1$ versus

$H_1$:$v{v}_1.$ The likelihood ratio is denoted by $.$ Is $-2log()$ asymptotically ${}_1^2-$distributed

under $H_0?$ If so$,$ why?