TITLE: Neyman-Pearson Theorem Question
QUESTION [0 upvotes]: Find the Neyman-Pearson test with size $\alpha$ to contrast
  $$ H_0: \beta = 1$$
$$ H_1: \beta = \beta_1$$
  with $\beta_1$ > 1 based in a sample of size 1 of the random variable with density:
  $$f(x,\beta) = \beta x^{\beta-1}, \ \ 0<x<1 $$

I've found that the rejection region is 
$$C = \{ x | \ \beta_1 x^{\beta_1-1} > k \} $$
I've been told that it's similar to 
$$C´ = \{ x | x < k´ \} $$ because C seems to decrease as x increase, but I don't understand why. Could you explain me why these two expressions are the same to solve the problem?

REPLY [1 votes]: The critical region you have found be NP Lemma, 
$$\frac{f_1(x)}{f_0(x)} \ge k$$
$$\implies x^{\beta_1-1} \ge k', \quad \beta_1 > 1$$
$$\implies x \ge k''$$
So we have that we'll reject $H_0$ if $x \ge k''$. 
The second critical region you have shown is not correct.
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