I meet a practical problem: How to do hypothesis test between two different distributions, for example:
$H_0$: distribution is a normal distribution $p(x) = \frac{1}{\sqrt{2\pi} \sigma}\exp(\frac{x^2}{2\sigma^2}) $
$H_1$: distribution is Laplace distribution $p(x) = \frac{1}{2b}\exp(-\frac{|x|}{b})$
with $\sigma$ and $b$ fixed.
How can I construct some statistics to do hypothesis test? How to construct uniformly most powerful(UMP) test?
Right now, I only know how to do hypothesis test of same kind distribution with different parameters. In general, what's the procedure to do hypothesis test between two different kinds of distribution and construct some meaningful statistics and UMP? Can you please provide some references, thanks.
If $\sigma$ and $b$ are fixed constants, we have two simple hypotheses.
The best test (formally: the most powerful) in this situation is given by Neyman–Pearson lemma.
The optimal criteria has the next form: $$S = \{ x: \frac{p_{H_1}(x)}{p_{H_0}(x)} \ge c\},$$
where $c$ is a solution of the equation $\int_{S} p_{H_0}(x)dx = \alpha$, where $\alpha$ is the significance level.
References: Testing Statistical Hypotheses, Lehmann, or any book about statistics.