P Values Maximized over a Confidence Set for the Nuisance Parameter
Journal article by Robert L. Berger, Dennis D. Boos; Journal of the American Statistical Association, Vol. 89, 1994
Journal Article Excerpt
P Values Maximized over a Confidence Set for the Nuisance Parameter. by Robert L. Berger , Dennis D. Boos 1. INTRODUCTION Testing problems are often complicated by the presence of a nuisance parameter vector [theta]. Consider first a model in which there is no nuisance parameter. Suppose that the data X have a probability distribution [P.sub.[nu]], defined in terms of a parameter [nu], and that we wish to test the simple hypothesis [H.sub.o: [nu] = [[nu].sub.o]. If the test statistic T is used to test [H.sub.o] and if large values of T give evidence against Ho, then for an observed value T = t, the p value is p = [P.sub.[nu]o] (T [greater than or equal to! t). Now consider a model with a nuisance parameter [theta]. The distribution of X has two parameters, [nu] and [theta]. We still wish to test [H.sub.o: [nu] = [[nu].sub.o], but this hypothesis is no longer simple, because the value of [theta] is unspecified. Using a test statistic as before, the p value is now p = [sup.sub.[theta]][P.sub.[nu]o,[delta]](T [greater than or equal to] t) (see, for example, Bickel and Doksum 1977, pp. 171-172). Unfortunately, the need to calculate the [sup.sub.[theta]] has complicated the problem. This complication is usually handled in one of three ways. First, in some problems it can be shown that for all values of t, the [sup.sub.[theta]] is always attained at a particular value [[theta].sub.o]. In this case the p value is simply p = [P.[nu]o,[delta]]o(T [greater than or equal to] t), and the parameter ([nu].sub.o] [[theta].sub.o]) is called the least favorable configuration. For example, in common one-sided testing problems, the boundary of the null hypothesis space is least favorable. A second way to handle the unknown [theta] is to choose judiciously a test statistic T (that usually depends on estimated values of [theta]) whose distribution under [H.sub.o] does not depend on [theta]. That is, T is ancillary under [H.sub.o]. Then [P.sub.[nu]o,[theta]](T [greater than or equal to] t) is the same for all [theta], so calculation of the sup,, is avoided. For example, in normal means problems we replace unknown variances with sample variances and use t or F distributions to account for the estimated variances. A third method to handle the unknown [theta] is to condition on the value of a statistic S that is sufficient for [theta] under [H.sub.o]. Then the conditional distribution of any statistic, given S, does not depend on [theta] (under [H.sub.o]), and the p value is taken to be p = [P.sub.[nu]o](T [greater than or equal to] t\S = s). For example, in a 2 X 2 contingency table with common "success" probability [theta] under [H.sub.o], one can condition on the marginals (a sufficient statistic for [theta] under [H.sub.o]) and use Fisher's exact test. All three methods replace the calculation of the sup, by the calculation of a single probability, and each method can result in a valid p value; that is, a statistic p such that under the null hypothesis, P(p [less than or equal to] [alpha]) [less than or equal to] [alpha], for each [alpha] [epsilon] [0, 1]. (1) We call a statistic that satisfies (1) a valid p value because it can be used in the standard way to define a level-[alpha] test. Consider the test that rejects the null hypothesis if and only if p [less than or equal to] [alpha]. Then under the null hypothesis, P(reject null) = P(p [less than or equal to] [alpha]) [less than or equal to! [alpha]; that is, the test so defined is a level-[alpha] test. In many situations, however, none of the three methods is satisfactory. For example, the value of [theta] at which the [sup.sub.[theta]] occurs may depend on the value t in a complicated way. Also, exact distributional results are often not available for statistics with estimated parameters. And, finally, it may not be possible to find an appropriate sufficient statistic on which to condition.In this article we consider a different approach for obtaining valid p values. Suppose that a valid p value p([[theta].sub.o]) may be calculated when the true value [[theta].sub.o] of the nuisance parameter vector [theta] is known. Here it should be noted that the calculation of p([[theta].sub.o]) does not have to be based on the same test statistic for different values of [[theta].sub.o]. Indeed the test statistic may depend directly on the assumed known value of [[theta].sub.o]. All that is needed is that for each value of [[theta].sub.o], p([[theta].sub.o]) must be a statistic that satisfies (1). If [[theta].sub.o] ... |
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