Access

You are not currently logged in.

Access JSTOR through your library or other institution:

If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Journal Article

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
Were these topics helpful?

Select the topics that are inaccurate.

Cancel
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Preview not available

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.

• 87
• 88
• 89
• 90
• 91
• 92
• 93