Randomized statistical test ($\varphi (x)\in [0,1]$)

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.