# example of uniformly most powerful test

let X be a single observation from the density $f(x;\theta)$ =$\theta x^{\theta -1} I_{(0,1)}(x)$

is there a UMP size-$\alpha$ test for testing $H_0 :\theta \ge \frac{1}{2}$ V/S $H_1 : \theta < \frac{1}{2}$ ?

Atempt :for general solution let $\theta_0 = \frac{1}{2}$

comparing $f(x;\theta)$ with the form $a(\theta)b(x)exp[c(\theta)d(x)]$ ,

we get $d(x) = log (x)$ and $c(\theta ) = \theta -1$ as monotone , increasing function in $\theta$ and therefor , critical region C =[$(x) :log(x) > k$ ] gives uniformly most powerful size - $\alpha$ test for testing

$H_0 :\theta \le \theta_0$ V/S $H_1 : \theta > \theta_0$

so for testing $H_0 :\theta \ge \theta_0$ V/S $H_1 : \theta < \theta_0$ , just the inequality in the critical region will get reversed i.e the critical region C =[$(x) :log(x) < k$ ] gives uniformly most powerful size - $\alpha$ test such that $P_{\theta_0}[log(x) < k$ ] = $\alpha$ .

is my logic correct ?

• Well, to finish the test maybe you should define constant $k_{\alpha}$ also Apr 5 '17 at 13:28
• @Lukasz_Grad that ill do , is the rest correct ? Apr 5 '17 at 13:49