The Annals of Mathematical Statistics

On the Fundamental Lemma of Neyman and Pearson

Abstract

The following lemma proved by Neyman and Pearson  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.

Article information

Source
Ann. Math. Statist., Volume 22, Number 1 (1951), 87-93.

Dates
First available in Project Euclid: 28 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177729695

Digital Object Identifier
doi:10.1214/aoms/1177729695

Mathematical Reviews number (MathSciNet)
MR39962

Zentralblatt MATH identifier
0042.14301

JSTOR