The Annals of Mathematical Statistics

A Generalization of the Neyman-Pearson Fundamental Lemma

Herman Chernoff and Henry Scheffe

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Given $m + n$ real integrable functions $f_1, \cdots, f_m, g_1, \cdots, g_n$ of a point $x$ in a Euclidean space $X$, a real function $\phi(z_1, \cdots, z_n)$ of $n$ real variables, and $m$ constants $c_1, \cdots, c_m$, the problem considered is the existence of a set $S^0$ in $X$ maximizing $\phi\big(\int_s g_1 dx, \cdots, \int_s g_n dx\big)$ subject to the $m$ side conditions $\int_s f_i dx = c_i$, and the derivation of necessary conditions and of sufficient conditions on $S^0$. In some applications the point with coordinates $\big(\int_s g_1 dx, \cdots, \int_s g_n dx\big)$ may also be required to lie in a given set. The results obtained are illustrated with an example of statistical interest. There is some discussion of the computational problem of finding the maximizing $S^0$.

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Ann. Math. Statist., Volume 23, Number 2 (1952), 213-225.

First available in Project Euclid: 28 April 2007

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Chernoff, Herman; Scheffe, Henry. A Generalization of the Neyman-Pearson Fundamental Lemma. Ann. Math. Statist. 23 (1952), no. 2, 213--225. doi:10.1214/aoms/1177729438.

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