Combining independent one-sample tests of significance Article Received: 09 November 1965
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Summary The problem of combining independent one-sample tests of significance is considered using techniques developed by the author (1965). LetX _{i1} ,..., X_{im} _{i} be positive observations andY _{i1} ,..., Y_{in} _{i} , the absolute values of negative observations in a sampleZ _{i1} ,..., Z_{iN} _{i} ofN _{i} =m_{i} +n_{i} independent and identically distributed random variables from a population with continuous cumulative distribution function\(\Pi _{0_i } (z);i = 1, \cdots ,k\) .

Then for testing the hypothesis that each of the distributions\(\Pi _{0_i } (z)\) is symmetric with respect to the origin, linear combinations of several one-sample test statistics are considered. Under suitable assumptions, two sets of combination coefficients are derived. One of them yields a class of tests with a region of consistency that is independent of the proportion of sample sizes (design-free tests) and the other has asymptotically the maximum power (locally asymptotically most powerful tests). Finally, these tests are compared with respect to the asymptotic values of their power against Pitman's shift alternatives and Lehmann's distribution free alternatives.

The research was sponsored by the Office of Naval Research under Contract Number Nonr-285 (38). Reproduction in whole or in part is permitted for any purpose of the United States Government.

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Authors and Affiliations 1. Courant Institute of Mathematical Sciences New York University New York USA