Parametric and permutation testing for multivariate monotonic alternatives

Abstract

We are firstly interested in testing the homogeneity of k mean vectors against two-sided restricted alternatives separately in multivariate normal distributions. This problem is a multivariate extension of Bartholomew (in Biometrica 46:328–335, 1959b) and an extension of Sasabuchi et al. (in Biometrica 70:465–472, 1983) and Kulatunga and Sasabuchi (in Mem. Fac. Sci., Kyushu Univ. Ser. A: Mathematica 38:151–161, 1984) to two-sided ordered hypotheses. We examine the problem of testing under two separate cases. One case is that covariance matrices are known, the other one is that covariance matrices are unknown but common. For the general case that covariance matrices are known the test statistic is obtained using the likelihood ratio method. When the known covariance matrices are common and diagonal, the null distribution of test statistic is derived and its critical values are computed at different significance levels. A Monte Carlo study is also presented to estimate the power of the test. A test statistic is proposed for the case when the common covariance matrices are unknown. Since it is difficult to compute the exact p-value for this problem of testing with the classical method when the covariance matrices are completely unknown, we first present a reformulation of the test statistic based on the orthogonal projections on the closed convex cones and then determine the upper bounds for its p-values. Also we provide a general nonparametric solution based on the permutation approach and nonparametric combination of dependent tests.

Appendix

Proof of lemma

(a) By defining the inner product, since M is a positive definite matrix, for A=(A ₁,…,A _k ) and B=(B ₁,…,B _k ), we have

Thus

and since for any closed convex cone $\mathcal{C}$, $M\pi_{S}(\mathbf{X},\mathcal{C}) = \pi_{S}(M\mathbf{X},M\mathcal{C})$, so we have that

(b) The proof is analogous to that in (a).

(c) To proof the part (c), first we show that

(6)

where

Suppose that ϑ _i and υ _i , i=1,2,…,k, be the estimators of the parameters on the closed convex cones $\mathcal{C}_{h}$ and $\mathcal{C}_{h}^{\prime}$, h=1,2,…,p, respectively. Then the right hand side of the formula (6) is

On the other hand it is known, see Lemma 1.1 in Zarantonello (1971), that $\mathbf{x}^{*} = \pi_{S}(\mathbf{x},\mathcal{C})$ if and only if $\mathbf{x}^{*} \in \mathcal{C}$, 〈x−x ^∗,x ^∗〉=0 and 〈x−x ^∗,B〉≤0 for all $B \in \mathcal{C}$.

Thus

So, we get that u≥0 and according to formula (6), we result that T _h ≥T(µ,µ ^∗), thus we complete the proof. □