# Combining independent one-sample tests of significance

## Summary

The problem of combining independent one-sample tests of significance is considered using techniques developed by the author (1965). Let*X* _{i1},..., X_{im} _{i} be positive observations and*Y* _{i1},..., Y_{in} _{i}, the absolute values of negative observations in a sample*Z* _{i1},..., Z_{iN} _{i} of*N* _{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.

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