Hypothesis Testing of a continuous distribution

Let $X_1, \dots, X_n$be a random sample of size $n$ from the continuous distribution with pdf given by

$f_x(x|\theta) = \theta x^{\theta - 1} I(x)_{(0,1)} I(\theta)$.

1) Show that $T = \prod^n_{i=1}X_i$ is sufficient for $\theta$.

2) Let $0 < \theta_0 < \theta_1$. Show that $\frac{f_x(x|\theta_1)}{f_x(x|\theta_0)}$ is an increasing function of $T$.

3) We are testing $H_0: \theta\geq 4$ vs. $H_A: \theta < 4$. Let $W = -\ln(T)$. Show that the test rejects $H_0$ if and only if $W>k$ is a UMP level $\alpha$ test where $\alpha = P(W>k|\theta = 4)$.

4) Derive the distribution for $W$.

5) Suppose that $n=14$. Use the appropriate tables to determine the level $\alpha$ of the UMP test from 3 when $k =4.0775$

1) Seems not too bad using the Factorization Theorem and setting that to be T and for 2), this seems live a lot will cancel and give me what I want when I plug in the quotient. I am pretty stuck with 3) and how to set that up but I think it is a form of a Beta distribution and I know I need to use Neyman-Pearson, I am just not quite sure how. 4) I think I have this by doing a transformation and getting an Exponential distribution, but not totally sure that is correct. 5) Not sure if this will be simpler once I get the others done or not.

Any help is greatly appreciated.

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I'm not sure what I have is correct, so I won't put down what I got so that I won't confuse you, when I'm already confused, but I will make a note for your part 3, which I'm sure is correct.

For part 3, what did you do to get the Beta distribution?

$\bf{Note:}$ The Neyman-Pearson Lemma can only be applied in a simple hypothesis test. Are your $H_{o}$ and $H_{A}$ simple hypotheses? If it's not a set of simple hypotheses, then is there another theorem that you can use that uses the fact that T is a sufficient statistic and that the MLR (from part 2) is non-decreasing? If you answered yes, then which theorem is it?

It's the Karlin-Rubin Theorem!

Here's where I found information on the Lemma. Lemma

For part 4, I also got the distribution is exponential, but then like I said before, I'm also not sure if I'm right on this question.

Good Luck with Part 5, I'm unsure as well.

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