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My Attempt

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

we get $d(x) = log (1-x)$ and $ c(\theta ) = \theta -1 $ as monotone , increasing function in $\theta$ and therefore , critical region C =[$(x) :log(1-x) > k$ ] 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 given hypothesis, my critical region is C =[$(x) :log(1-x) > k$ ] and then we can find that answer will be option A.

Is my attempt correct?


1 Answer 1


Your attempt is correct, but for this particular multiple choice question we can find the cut-off point $c$ directly given that the critical region is of the form $-\sum\limits_{i=1}^n \ln(1-X_i)^2 <c$.

As you might have already noticed, $$X_i\sim \mathsf{Beta}(1,\theta)\implies 1-X_i\sim \mathsf{Beta}(\theta,1)\implies-2\theta\ln (1-X_i)\sim \chi^2_2$$

Since size of the test is $\alpha$, we have

$$P_{H_0}\left[-\sum_{i=1}^n \ln(1-X_i)^2 <c\right]=P_{H_0}\left[-2\sum_{i=1}^n \ln(1-X_i)< c\right]=P\left[\chi^2_{2n}< c\right]=\alpha$$

Or, $$P\left[\chi^2_{2n}> c\right]=1-\alpha\quad,\,\text{i.e.}\quad c=\chi^2_{2n,1-\alpha}$$


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