Showing papers by "Herman Chernoff published in 1952"

A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations

Abstract: In many cases an optimum or computationally convenient test of a simple hypothesis $H_0$ against a simple alternative $H_1$ may be given in the following form. Reject $H_0$ if $S_n = \sum^n_{j=1} X_j \leqq k,$ where $X_1, X_2, \cdots, X_n$ are $n$ independent observations of a chance variable $X$ whose distribution depends on the true hypothesis and where $k$ is some appropriate number. In particular the likelihood ratio test for fixed sample size can be reduced to this form. It is shown that with each test of the above form there is associated an index $\rho$. If $\rho_1$ and $\rho_2$ are the indices corresponding to two alternative tests $e = \log \rho_1/\log \rho_2$ measures the relative efficiency of these tests in the following sense. For large samples, a sample of size $n$ with the first test will give about the same probabilities of error as a sample of size $en$ with the second test. To obtain the above result, use is made of the fact that $P(S_n \leqq na)$ behaves roughly like $m^n$ where $m$ is the minimum value assumed by the moment generating function of $X - a$. It is shown that if $H_0$ and $H_1$ specify probability distributions of $X$ which are very close to each other, one may approximate $\rho$ by assuming that $X$ is normally distributed.

A Generalization of the Neyman-Pearson Fundamental Lemma

Abstract: 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 $S^0$.