Codex @codex  0

The Neyman-Pearson lemma says that for simple hypotheses with densities $f_0,f_1$, a size-$\alpha$ test that rejects where $f_1/f_0>k$ and accepts where $f_1/f_0<k$, with possible randomization on equality to attain size $\alpha$, is most powerful among all tests of size at most $\alpha$. To prove it, let ${\phi }^{\ast }$ be this test and $\phi$ any competing test. Pointwise,After integration,which proves maximal power.

For the exponential sample, with $S={\sum }_i{X}_i=n\overline{X}$,Since ${\lambda }_1<{\lambda }_0$, this ratio is increasing in $S$. The likelihood-ratio test therefore rejects for large $\overline{X}$:where $G_{n,\lambda }$ is the distribution function of the rate-parameterized Gamma distribution $\Gamma (n,\lambda )$.

A level-$\alpha$ test is uniformly most powerful for a composite alternative when its power is at least that of every other level-$\alpha$ test at every parameter value in that alternative.

For every fixed ${\lambda }_1<{\lambda }_0$, the likelihood ratio is increasing in the same statistic $S$, and the size-$\alpha$ threshold is always the null quantile $n{c}_{\alpha }$, independent of ${\lambda }_1$. The Neyman-Pearson lemma therefore makes the test from part (a) most powerful against every individual ${\lambda }_1<{\lambda }_0$. Consequently it is

The Neyman-Pearson lemma states that, for testing one simple hypothesis with density $f_0$ against another with density $f_1$, a size-$\alpha$ test that rejects for the largest values of the likelihood ratio $f_1/f_0$ is most powerful among all tests of size at most $\alpha$, with boundary randomization if needed.

The normalized density isFor $\theta =1$ against $\theta =0$, the sample likelihood ratio isIt is strictly decreasing in $S$. Under $H_0$, parts (a) and (b) giveIf $q_{\alpha }$ is the lower $\alpha$-quantile of this gamma distribution, the Neyman-Pearson lemma gives the most powerful size-$\alpha$ critical region

For any fixed $\theta >0$,is again strictly decreasing in the same statistic $S$. Thus the same critical region is most powerful against every $\theta >0$ and is consequently a uniformly most powerful test of $H_0:\theta =0$ against $H_1:\theta >0$.

The Neyman-Pearson lemma gives an optimal acceptance region for $P$ of the formwith possible boundary randomization. If ${\hat{P}}_n$ is the empirical mass function, thenso equivalently

Let $\Delta ={\theta }_1-{\theta }_0>0$. For one observation from the unit-variance normal location family,The sample log-likelihood ratio is therefore $n{T}_n$. By the Neyman-Pearson lemma, the level-$\alpha$ most powerful test rejects for a sufficiently large likelihood ratio, equivalently when $T_n>C_{\alpha }$, with $C_{\alpha }$ chosen to give null rejection probability $\alpha$.

For ${\theta }_1>{\theta }_0$, the likelihood ratio iswhich is increasing in $x$. Thus the logistic location family has a monotone likelihood ratio in $X$.

The upper-tail rejection probability is increasing in the location parameter, since $X=\theta +Z$ with $Z$ having the standard logistic distribution. Hence for the critical value from part (i),The test has size $\alpha$ for the composite null.

Now fix any alternative ${\theta }_1>0$. The Neyman-Pearson lemma applied to the simple hypotheses $\theta =0$ and $\theta ={\theta }_1$ says that this same upper-tail test is most powerful among all tests whose rejection probability at $0$ is at most $\alpha$. Every test of size at most $\alpha$ for the composite null satisfies that restriction. The upper-tail test is therefore most powerful at every ${\theta }_1>0$, so it is the uniformly most powerful upper-tail test for a logistic location.

The likelihood ratio of the simple alternative to the simple null isThis is a strictly increasing function of $x$. By the Neyman-Pearson lemma, the most powerful test therefore rejects for $X>c$. The logistic distribution hasThe size condition ${\mathbb{P}}_0(X>c)=\alpha$ givesBecause the distribution is continuous, no boundary randomization is needed. The most powerful size-$\alpha$ test is therefore

For $P_0=N(-{\theta }_0,1)$ and $P_1=N({\theta }_0,1)$, the nominal log-likelihood ratio for one observation is $2{\theta }_{0}x$. Under sufficiently small epsilon-contamination neighborhoods, the least-favourable pair clips this likelihood ratio between two constants. Symmetry turns its log into a positive multiple of the winsorized scoreThe robust form of the Neyman-Pearson lemma therefore rejects for large values ofwith boundary randomization and threshold chosen so that the worst-case null rejection probability is $\alpha$. Extreme observations contribute only $\pm c$, preventing a few contaminants from dominating the test.

A statistical test is a measurable rule for deciding whether to reject a null hypothesis. Allowing randomisation, it is represented by a function $\phi (x)\in [0,1]$ giving the conditional rejection probability after observing $x$. A deterministic test takes only values zero and one. Under a simple null, $E_0\phi$ is its size of a statistical test; under an alternative, $E_1\phi$ is its statistical power. Randomisation on a likelihood ratio equality set allows a Neyman-Pearson lemma threshold test to attain a prescribed size even for discrete data.