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# On the Fundamental Lemma of Neyman and Pearson

George B. Dantzig and Abraham Wald
The Annals of Mathematical Statistics
Vol. 22, No. 1 (Mar., 1951), pp. 87-93
Stable URL: http://www.jstor.org/stable/2236704
Page Count: 7
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## Abstract

The following lemma proved by Neyman and Pearson [1] is basic in the theory of testing statistical hypotheses: LEMMA. Let f1(x), ⋯, fm+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, c1, ⋯, cm be m given constants and S the class of all Borel measurable subsets S of R for which (1.1) ∫S fi(x) dx = ci (i = 1, ⋯, m). Let, finally, S0 be the subclass of S consisting of all members S0 of S for which (1.2) ∫S0 fm + 1(x) dx ≥ ∫S fm+1(x) dx for all S in S. If S is a member of S and if there exist m constants k1, ⋯, km such that (1.3) fm + 1(x) ≥ k1f 1(x) + ⋯ + kmf m(x) when x ε 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 S0. The above lemma gives merely a sufficient condition for a member S of S to be also a member of S0. Two important questions were left open by Neyman and Pearson: (1) the question of existence, that is, the question whether S0 is non-empty whenever 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 S0 is not empty whenever S is not empty. In Section 3, a necessary and sufficient condition is given for a member of S to be also a member of S0. This necessary and sufficient condition coincides with the Neyman-Pearson sufficient condition under a mild restriction.

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