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I have a discrete random variable $N$, from which a random sample $N_1,\dots,N_{15}$ is drawn. I want to test two hypotheses about the distribution of $N$ which are represented by two histograms. In other words, the two hypotheses are$$\begin{align}H_0:\mathbb P(N\in [x_i,x_{i+1}]|H_0)&=p_i\\H_1:\mathbb P(N\in [x_i,x_{i+1}]|H_1)&=q_i\end{align}$$where $[x_i,x_{i+1}]$ are the histogram bins, and $p_i$ and $q_i$ are known.

I thought I can use the Neyman-Pearson lemma, by using $$\Lambda(\mathbf n) = \prod_{i=0}^{15} \frac{\mathbb P(N_i=n_i|H_0)}{\mathbb P(N_i=n_i|H_1)}$$ as a test statistic, but I feel I need to calculate its distribution in order to evaluate the significance of the test $$\alpha = \mathbb P (\Lambda \le k_\alpha | H_0) $$ and I have no idea how to calculate $k_\alpha$ for a given $\alpha$ (or even vice versa).

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It seems like there is no way to rewrite the equation $$\alpha = \mathbb P (\Lambda \le k_\alpha | H_0)$$ in any way. The only way of estimating the RHS is through Monte Carlo techniques.

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