Question

Suppose Xi and X2 are iid from 0, otherwise, where θ 0, and consider testing Ho : θ 1 versus H1 : θ 1 . We have two tests: where 0<c<1 (a) Show that the power functions of the two tests are A(0)-1-(0.9)θ and β2(0)-1 + d|θ Inc-1), respectively. (b) Calculate the size of the φι test. Then, find the value of c that gives the same size for the φ2 test. (c) Is фг a most powerful test of HÓ : θ = 1 versus Hl : θ = 2? Explain.

Answer 1

$(a)~\beta_1(\theta)=P(X_1>0.9|\theta)=\int_{0.9}^1\theta x^{\theta -1}dx=1-(0.9)^\theta.\\ Transform~X\rightarrow Y~where\\ y=-2\theta \log x;~x=e^{-\frac{y}{2\theta}};~|Jacobian|=|\frac{\mathrm{d} x}{\mathrm{d} y}|=\frac{1}{2\theta}e^{-\frac{y}{2\theta}};~0<x<\Rightarrow y>0\\ The~pdf~of~Y~is\\ f_Y(y)=\frac{1}{2}e^{-\frac{y}{2}};~y>0\\ =0~otherwise\\ -2\theta \log X\sim \chi^2_2\\ \beta_2(\theta)=P(X_1X_2>c|\theta)=P(-2\theta\log X_1-2\theta\log X_2<-2\theta\log c|\theta)\\ =\int_0^{-2\theta \log c}\frac{1}{4}e^{-\frac{y}{2}}ydy\\ ~\left[since~X_1~and~X_2~are~independent~then~-2\theta\log X_1-2\theta\log X_2\sim \chi^2_4 \right ]\\ =\frac{1}{4}\left[-2ye^{-\frac{y}{2}}-4e^{-\frac{y}{2}} \right ]^{-2\theta\log c}_0=1&plus;c^\theta(\theta\log c-1).\\ (b)~Size~of~\phi_1~test=\beta_1(1)=1-0.9=0.1\\ \beta_2(1)=0.1\\ or,~1&plus;c(\log c-1)=0.1~or,~c\log c-c=0.9~or,~c=2.7183~\\(using~iteration~method~and~initial~value=2.7).\\$

$(c)~f_{H_1}(x_1,x_2)>kf_{H_0}(x_1,x_2)\\ or,~2^2(x_1x_2)>k~or,~x_1x_2>c\\ Hence~most~powerful~critical~region~is~\{(x_1,x_2):x_1x_2>c\}~which~is~same~as\\~critical~region~for~\phi_2~test,~ hence~\phi_2~is~a~most~powerful~test~of\\~H_0^\prime:\theta=1~vs.~H_1^\prime:\theta=2.$