On the Fundamental Lemma of Neyman and Pearson

Abstract

The following lemma proved by Neyman and Pearson [1] is basic in the theory of testing statistical hypotheses: LEMMA. Let $f_1(x), \cdots, f_{m+1}(x)$ be $m + 1$ Borel measurable functions defined over a finite dimensional Euclidean space $R$ such that $\int_R |f_i(x)|dx < \infty (i = 1, \cdots, m + 1)$. Let, furthermore, $c_1, \cdots, c_m$ be $m$ given constants and $\mathcal{S}$ the class of all Borel measurable subsets $S$ of $R$ for which (1.1) $\int_S f_i(x) dx = c_i \\ (i = 1, \cdots, m)$. Let, finally, $\mathcal{S}_0$ be the subclass of $\mathcal{S}$ consisting of all members $\mathcal{S}_0$ of $\mathcal{S}$ for which (1.2) $\int_{S_0} f_{m + 1}(x) dx \geqq \int_S f_{m+1}(x) dx \text{for all S in} \mathcal{S}$. If $S$ is a member of $\mathcal{S}$ and if there exist $m$ constants $k_1, \cdots, k_m$ such that (1.3) $f_{m + 1}(x) \geqq k_1f_1(x) + \cdots + k_mf_m(x) \text{when} x \epsilon S$, (1.4) $f_{m + 1}(x) \leqq k_1f_1(x) + \cdots + k_mf_m(x) \text{when} x \not\epsilon S$, then $S$ is a member of $\mathcal{S}_0$. The above lemma gives merely a sufficient condition for a member $S$ of $\mathcal{S}$ to be also a member of $\mathcal{S}_0$. Two important questions were left open by Neyman and Pearson: (1) the question of existence, that is, the question whether $\mathcal{S}_0$ is non-empty whenever $\mathcal{S}$ is non-empty; (2) the question of necessity of their sufficient condition (apart from the obvious weakening that (1.3) and (1.4) may be violated on a set of measure zero). The purpose of the present note is to answer the above two questions. It will be shown in Section 2 that $\mathcal{S}_0$ is not empty whenever $\mathcal{S}$ is not empty. In Section 3, a necessary and sufficient condition is given for a member of $\mathcal{S}$ to be also a member of $\mathcal{S}_0$. This necessary and sufficient condition coincides with the Neyman-Pearson sufficient condition under a mild restriction.