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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 ?

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  • $\begingroup$ Well, to finish the test maybe you should define constant $k_{\alpha}$ also $\endgroup$
    – Łukasz Grad
    Commented Apr 5, 2017 at 13:28
  • $\begingroup$ @Lukasz_Grad that ill do , is the rest correct ? $\endgroup$
    – ANUJ NAIN
    Commented Apr 5, 2017 at 13:49

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