Showing papers by "Herman Chernoff published in 1958"

Asymptotic normality and efficiency of certain nonparametric test statistics

Abstract: Theorems of Wald and Wolfowitz, Noether, Hoeffding, Lehmann, Madow, and Dwass have given sufficient conditions for the asymptotic normality of TN. In this paper we extend some of these results to cover more situations with F s G. In particular it is shown for all alternative hypotheses that the FisherYates-Terry-Hoeffding cl-statistic is asymptotically normal and the test for translation based on it is at least as efficient as the t-test. 2. Introduction. Finding the distributions of nonparametric test statistics and establishing optimum properties of these tests for small samples has progressed slower than the corresponding large sample theory. Even so, it is not possible to state that the basic framework of the large sample theory has been completed. Dwass [3] has recently presented a general theorem on the asymptotic normality of certain nonparametric test statistics under alternative hypotheses. His results, however, do not apply to such important and interesting procedures as the cl-test [11]. Many papers have appeared giving the asymptotic efficiency of particular tests. Hodges and Lehmann [7] have discussed the asymptotic efficiency of the Wilcoxon test with respect to all translation alternatives. In the same paper they have conjectured that the cl-test is as efficient as the ttest for normal alternatives and at least as efficient as the t-test for all other alternatives. The beginning of our work came from a desire to verify the Hodges and Lehmann conjecture. Related to the conjecture is the hypothesis that the cl-statistic is asymptotically normally distributed. Thus our work has two parts: developing a new theorem for asymptotic normality of nonparametric test statistics and the establishing of the variational argument required for determining the minimum efficiency of test procedures.

Central Limit Theorems for Interchangeable Processes

Abstract: Let {Xn} (n = 1, 2 , …) be a stochastic process. The random variables comprising it or the process itself will be said to be interchangeable if, for any choice of distinct positive integers i 1, i 2, H 3 … , ik, the joint distribution of depends merely on k and is independent of the integers i 1, i 2, … , i k. It was shown by De Finetti (3) that the probability measure for any interchangeable process is a mixture of probability measures of processes each consisting of independent and identically distributed random variables.