Two-Stage Likelihood Ratio and Union–Intersection Tests for One-Sided Alternatives Multivariate Mean with Nuisance Dispersion Matrix

Abstract

For a multinormal distribution with an unknown dispersion matrix, union-intersection (UI) tests for the mean against one-sided alternatives are considered. The null distribution of the UI test statistic is derived and its power monotonicity properties are studied. A Stain-type two-stage procedure is proposed to eliminate some of the inherent drawbacks of such tests. Some comparisons are also made with some recently proposed alternative conditional likelihood ratio tests. Academic PressAMS 1991 subject classifications: 62H15, 62L10.

A Comparison of Aggregate P-Value Methods and Multivariate Statistics for Self-Contained Tests of Metabolic Pathway Analysis

“…For one-sided alternatives, we propose a simple modification of statistics of the form (2.4) given in Sen [ 11 ], which was used in the context of union-intersection tests for one-sample problems: where and S are the usual one-sample mean vector and covariance matrix for testing a series of hypotheses of the form H 0, b = b′ θ = 0. We propose the natural two-sample analog.…”

Asymptotically optimal tests for parametric functions against ordered functional alternatives

“…Therefore, the UIT proposed in Section 3, as modified forT n(1) , is asymptotically power-equivalent to the corresponding marginal likelihood ratio test, though the latter may be harder to formulate. This may be illustrated with Example 5 where and are orthogonal, though the Barndorff-Nielsen characterization or the factorization of the likelihood function may not hold; we refer to Sen and Tsai [26] for the asymptotic power-equivalence of UIT and Perlman's [18] LRT for this specific problem.…”

“…Even for k = 2, there are some difficulties for drawing exact inference and the situation becomes worse for k 3 (Glesar and Hwang [11]). [29], Sen and Tsai [26], and Perlman and Wu [19]). Still, the finite sample resolutions are not final say in this matter.…”

A generalized likelihood ratio test for normal mean when p is greater than n

“…The union-intersection method of test construction might be useful when the null hypothesis is conveniently expressed as an intersection. For more references about the union-intersection test, see, for example, Roy (1954), Mudholkar et al (1974), Olkin and Tomsky (1981), Cohen et al (1994), Sen and Tsai (1999). The UIT for the hypothesis (1) can be constructed as follow.…”