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# Generalized p-Values in Significance Testing of Hypotheses in the Presence of Nuisance Parameters

Journal of the American Statistical Association
Vol. 84, No. 406 (Jun., 1989), pp. 602-607
DOI: 10.2307/2289949
Stable URL: http://www.jstor.org/stable/2289949
Page Count: 6
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## Abstract

This article examines some problems of significance testing for one-sided hypotheses of the form H0: θ ≤ θ0 versus $H_1: \theta > \theta_0$, where θ is the parameter of interest. In the usual setting, let x be the observed data and let T(X) be a test statistic such that the family of distributions of T(X) is stochastically increasing in θ. Define Cx as {X: T(X) - T(x) ≤ 0}. The p value is $p(x) = \sup_{\theta \leq \theta_0} \Pr(X \in C_x \mid \theta)$. In the presence of a nuisance parameter η, there may not exist a nontrivial Cx with a p value independent of η. We consider tests based on generalized extreme regions of the form Cx(θ, η) = {X: T(X; x, θ, η) ≥ T(x; x, θ, η)}, and conditions on T(X, x, θ, η) are given such that the p value $p(x) = \sup_{\theta \leq \theta_0} Pr(X \in C_x \mid \theta)$ is free of the nuisance parameter η, where T is stochastically increasing in θ. We provide a solution to the problem of testing hypotheses about the differences in means of two independent exponential distributions, a problem for which the fixed-level testing approach has not produced a nontrivial solution except in a special case. We also provide an exact solution to the Behrens-Fisher problem. The p value for the Behrens-Fisher problem turns out to be numerically (but not logically) the same as Jeffreys's Bayesian solution and the Behrens-Fisher fiducial solution. Our approach of testing on the basis of p values is especially useful in multiparameter problems where nontrivial tests with a fixed level of significance are difficult or impossible to obtain.

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