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EXACT P ‐VALUES FOR DISCRETE MODELS OBTAINED BY ESTIMATION AND MAXIMIZATION
Author(s) -
Lloyd Chris J.
Publication year - 2008
Publication title -
australian and new zealand journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 1369-1473
DOI - 10.1111/j.1467-842x.2008.00520.x
Subject(s) - maximization , mathematics , nuisance parameter , value (mathematics) , range (aeronautics) , statistic , expectation–maximization algorithm , set (abstract data type) , test statistic , mathematical optimization , statistics , statistical hypothesis testing , maximum likelihood , computer science , materials science , estimator , composite material , programming language
Summary In constructing exact tests from discrete data, one must deal with the possible dependence of the P ‐value on nuisance parameter(s) ψ as well as the discreteness of the sample space. A classical but heavy‐handed approach is to maximize over ψ. We prove what has previously been understood informally, namely that maximization produces the unique and smallest possible P ‐value subject to the ordering induced by the underlying test statistic and test validity. On the other hand, allowing for the worst case will be more attractive when the P ‐value is less dependent on ψ. We investigate the extent to which estimating ψ under the null reduces this dependence. An approach somewhere between full maximization and estimation is partial maximization, with appropriate penalty, as introduced by Berger & Boos (1994, P values maximized over a confidence set for the nuisance parameter. J. Amer. Statist. Assoc. 89 , 1012–1016). It is argued that estimation followed by maximization is an attractive, but computationally more demanding, alternative to partial maximization. We illustrate the ideas on a range of low‐dimensional but important examples for which the alternative methods can be investigated completely numerically.