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Showing papers by "Herman Chernoff published in 1951"


Journal ArticleDOI
01 May 1951
TL;DR: In this paper, the authors extend this result to the non-convex case and show that the range of a countably additive finite measure with values in a finite-dimensional real vector space is bounded and closed.
Abstract: Liapounoff2 established in 1940 that the range of a countably additive finite measure with values in a finite-dimensional real vector space is bounded and closed and in the nonatomic case convex. A simplified proof of this result was given by Halmos' in 1948. The aim of the present paper is to extend this result to the following case. Let pit, 1 0 there exist a a>0 such that g*(E)

9 citations


Journal ArticleDOI
TL;DR: In a test of an hypothesis, one may regard a sample in the critical region as evidence that the hypothesis is false as mentioned in this paper, and for some reason it is desired to increase the critical size of the test, i.e., to make rejection of the hypothesis more probable.
Abstract: In a test of an hypothesis one may regard a sample in the critical region as evidence that the hypothesis is false Let us assume that for some reason it is desired to increase the critical size of the test, ie, to make rejection of the hypothesis more probable Then one may expect that an observation which led to rejection in the first test should still lead to rejection in the new test In other words, one should expect $W_\alpha \supset W_{\alpha'}$ if $\alpha > \alpha'$, where $W_\alpha$ is the critical region for the test of size $\alpha$ An example is given where regions of type $A$ are uniquely specified except for sets of measure zero, but fail to have this property

7 citations