Most powerful size $\alpha$ test using a single observation

Question

I am trying to find the most powerful size $\alpha$ test for a single observation

$$X\sim f(X;\theta)=(2\theta X+1-\theta)I(0<X<1)\,,$$

where $\theta\in[-1,1]$, for the hypothesis $H_o:\theta=0$ and $H_a:\theta=1$

The likelihood ratio:

$$\lambda(X)=\frac{f(X;\theta=0)}{f(X;\theta=1)}=\frac{1}{2X}$$

Note want to reject for small value of $\frac{1}{2X}$ that is large values of $X$ to find the rejection region $\lambda(X)\leq c$ can equivalently find the rejection region $X\ge c$. Consider:

$$\alpha=P(X\ge c;\theta=0)=\int_{c}^{1}1\,dx=1-c$$ then $c=1-\alpha$ will produce a $\alpha$ level test.

For the power:

$$P(X\ge c; \theta=1)=\int_{c}^{1}2x \,dx=1-c^2$$ to maximize power want to make $c$ as small as possible to maximize power but since only option is $c=1-\alpha$ and since any $c\ge 1-\alpha$ will produce an $\alpha$ level size test then $c=1-\alpha$ being the smallest amongst this group, rejecting when:

$$\frac{1}{2x}\le 1-\alpha $$

is the most powerful $\alpha$ size test. Is this correct?

Answer 1

All was well done up to the line "then $c=1−\alpha$ will produce a $\alpha$ level test."

The test that you have constructed is already the most powerful $\alpha$-level test according to Neyman-Pearson Lemma.

It seems to me that you do not quite understand the definition. There are many tests for these two hypotheses whose level equal to or does not exceed $\alpha$.

For example, the test with a critical region $X<\alpha$ also has an $\alpha$ level.

The test with a critical region $X\in[0,\frac{\alpha}{2})\cup(1-\frac{\alpha}2,1]$ also has an $\alpha$ level. And so on.

But among all of them, the likelihood ratio criterion of $\alpha$ level has the greatest power. This is the statement of the Lemma.

So to construct MP $\alpha$-level test you need to constuct a likelihood ratio test and equate its level to $\alpha$, which exactly you did.

You can in addition calculate its power $\beta=1-(1-\alpha)^2$ but it is not needed to do smth with it. You can admire this value. You can say that here it is - the biggest power of the test of level $\alpha$, more than which no test of level $\alpha$ can ever have. But hardly anything else.