This is a homework question. I was given a random sample of independent and identically distributed $X_i$'s and wish to test the hypotheses:
$$H_0: \theta = \theta_0$$
$\text{vs}$
$$H_A: \theta = \theta_A \quad (\theta_A < \theta_0)$$
$f(x;\theta)=0.5(1+\theta \, x)\,$ where $-1\leq \theta \leq 1$ and $-1\leq x \leq1$
In this case, I should apply the Neyman Pearson Lemma, as we are testing for a simple null and simple alternate
However, the question asked to show that there is no most powerful test in this case, and explain why
What I did was applying the N-P Lemma, so
NP: $[(\frac{1}{2^n})\,\prod (1+\theta_A \, x_i)]/[(\frac{1}{2^n})\, \prod (1+\theta_0 \, x_i)]>k$
and if i take the log of it i get
$\sum[\log((1+\theta_A\, x_i)/(1+\theta_0 \, x_i))]\geq \ln k$
And then I concluded, since there is no simple closed way to arrange the LHS to be in the form of a test statistic, of only the data, and the test statistic depends on $\theta_A$, therefore there is no most powerful test in this case, is this a correct assessment of the situation?
Also, please correct my misunderstanding, I was under the assumption that if we are testing for simple null and simple alternate, there is always a most powerful test as the Neyman-Pearson lemma suggests, as evident by this question, my understanding was wrong, but I am not sure in what way does the Neyman-Pearson lemma give the most powerful test?