Suppose that we have a random sample $X_1,X_2,...,X_n$ from a prob function with density: $$ f(x) = 3\theta^3x^{-4} $$ given that $x\geq \theta$
Now the question is
use Neyman-Pearson's Lemma to find a most powerful test for testing $H_0:\theta =4$ against $H_a : \theta = 3$
Now I keep getting into trouble with Neyman-Pearson. So If anybody could give me a hint as to what I am doing wrong, that'd be great.
The combined PDF is defined as: $$ f_\bar{x}(x_1,\ldots,x_n)=(3\theta^3)^n \prod_{i=1}^nx_i^{-4}I_{(\theta,\infty)}(x_i) $$ As far as I know. Now implementing this witht he two thetas I get: $$ \begin{equation} \lambda(x_1,\dots,x_n)=\frac{(3\times4^3)^n \prod_{i=1}^nx_i^{-4}I_{(4,\infty)}(x_i)}{(3\times 3^3)^n \prod_{i=1}^nx_i^{-4}I_{(3,\infty)}(x_i)} \tag{1} \end{equation} $$
Now my book says the formula is $$ \lambda(x_1,\dots,x_n)=\dfrac{f(x_1,\dots,x_n;\theta_0)}{f(x_1,\dots,x_n;\theta_a)} $$ and they say let $C^*$ be the set $$ C^* = \{(x_1,\dots,x_n)|\lambda (x_1,\dots,x_n;\theta_0,\theta_a)\leq k\} $$ So essentially they're saying, $C^*$ is the critical region for which the $\lambda$ fraction is less than a certain $k$, for which k is defined as a constant such that: $$ P[(X_1,\ldots,X_n)\in C^*|\theta_0]=\alpha $$
Now as was suggested I drew $(1)$. for $n=1$, so taking $f_0(x)=81x^{-4}I_{(3,\infty)}(x)$ and $f_1(x)=3\times 64x^{-4}I_{(4,\infty)}(x)$ we get: $$ \dfrac{81x^{-4}}{3\times 64x^{-4}}I_{(4,\infty)}(x) $$ However, because the $x^{-4}$ cancel out I just get $\dfrac{27}{64}$. so taking the integral from $4 \to \infty$ gives me $\infty$.
In the general case for $n\to\infty$ $$ L(\theta) = \infty I_{(3,4)} + \underbrace{ \frac{27}{64} I_{(4,\infty)}}_{\text{what does this tell me?}} $$
Hint: First of all to calculate the likehood funtion, you have multiplication not the summation. Second $I(\theta,\infty)$ in your last formula is incorret. In the nominator it should read $I(\theta_1,\infty)$ and in the denominator $I(\theta_0,\infty)$, where $\theta_0$ and $\theta_1$ are $4$ and $3$ respectively.
Then take for simplicity, $n=1$. Now you have two simple hypothesis. Answer the following questions: "what decision should you give if $3<x<4$" and where should you put the threshold to get a specific false alarm rate?.
Then let $n\neq 1$, how does your densities look like and eventually what is the distribution of your likelihood function under the $H_0$ and $H_1$. From there you need to integrate the distribution of the likelihood under $H_0$ from some thereshold to $\infty$ which will give you some $\alpha$, that is your false alarm rate. This threshold will also maximize your detection rate.