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The Neyman-Pearson lemma concerns a simple hypothesis $H_0:\theta ={\theta }_0$ against another simple hypothesis $H_1:\theta ={\theta }_1$. For a continuous statistical sample, put $f_j(x)={\prod }_{i}f(x_i;{\theta }_j)$ for the joint probability density functions, $j=0,1$. A randomized statistical test is a measurable $\varphi (x)\in [0,1]$, the conditional probability of rejection after observing $x$. Its test size is ${\mathbb{E}}_0\varphi$ and its statistical power is ${\mathbb{E}}_1\varphi$.

Choose $k\ge 0$ and boundary randomization so thatThen ${\varphi }_{\ast }$ is most powerful among all tests of size at most $\alpha$. Thus one rejects for sufficiently large alternative-to-null likelihood ratio. If the equality set has zero probability under $H_0$, the test needs no randomization. Continuous observations alone do not guarantee that their likelihood ratio has a continuous probability distribution.

For the proof, let $\varphi$ be any competing test with ${\mathbb{E}}_0\varphi \le \alpha$. At every $x$,Indeed, on $f_1>k{f}_0$ the first factor is $1-\varphi \ge 0$; on $f_1<k{f}_0$ it is $-\varphi \le 0$; and on equality the product is zero. Integration giveswhich proves the claimed optimality. Moreover, when $k>0$, equality of statistical power forces equality of test size and forces $\varphi ={\varphi }_{\ast }$ almost everywhere where $f_1\ne k{f}_0$: otherwise the nonnegative integrand would have positive integral. This proves the usual necessity assertion as well, up to equality sets and sets of zero probability under both hypotheses.

A randomized statistical test specifies a measurable rejection probability $\varphi (x)$ for each possible data value. It can be implemented by rejecting when an independent uniform random variable $U$ on $(0,1)$ satisfies $U<\varphi (x)$. A deterministic test has only the values zero and one. Under a simple hypothesis $H_j$, its rejection probability is ${\mathbb{E}}_j\varphi (X)$, which is its test size for $j=0$ and its statistical power for $j=1$.

The Neyman-Pearson lemma permits randomization on an equality set of the form $f_1(x)=k{f}_0(x)$ to reach a prescribed test size. Continuous observations do not rule out an equality set of positive probability: if both hypotheses give the same probability density function, their likelihood ratio is identically one. This explains why a precise statement of the lemma includes boundary randomization even for continuous data.