Let X1,X2,…,Xn be a random sample from a distribution with probability density function ftheta(x)=begincasestheta(1-x)^theta-1,&0<x<1,0,&otherwise,endcasesqquadtheta0 To test H0:theta=1 against H1:theta1, the uniformly most powerful test of size alpha (0<alpha<1) would reject H0 if

(A) -sumi=1^nloge(1-Xi)^2<chi^22n,1-alpha

Q30Section A (Section A)30 of 60

UMP test for a Beta family

Let $X_{1},X_{2},…,X_{n}$ be a random sample from a distribution with probability density function $f_{\theta}(x)=\begin{cases}\theta(1-x)^{\theta-1},&0<x<1,\\0,&\text{otherwise},\end{cases}\qquad\theta>0$ To test $H_{0}:\theta=1$ against $H_{1}:\theta>1$ , the uniformly most powerful test of size $\alpha$ ( $0<\alpha<1$ ) would reject $H_{0}$ if

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