On the power of Wilks' U-test for MANOVA

Let V₁,…, V_m, W₁,…, W_n be independent p × 1 random vectors having multivariate normal distributions with common nonsingular covariance matrix Σ and with EW_α = 0, α = 1,…, n. In this canonical form of the multivariate linear model, the problem is to test H: EV_αazμ_α = 0, α = 1,…, m vs K: not H. It is shown that when the rank of the noncentrality matrix (μ₁…μ_m) Σ^?1 (μ₁…μ_m) is one, the power of Wilks' U-test (the likelihood ratio test) strictly decreases with the dimension p and the hypothesis degrees of freedom m. This generalizes results known for the noncentral F-test in the univariate case.     

On the asymptotic joint distributions of certain functions of the eigenvalues of four random matrices

In this paper, the authors obtained asymptotic expressions for the joint distributions of certain functions of the eigenvalues of the Wishart matrix, correlation matrix, MANOVA matrix and canonical correlation matrix when the population roots have multiplicity.     

On the asymptotic power of some k-sample statistics based on the multivariate empirical process

Some k-sample Kolmogorov-Smirnov and Cramér-von Mises-type statistics, based on the multivariate empirical process, are studied. Expressions for their asymptotic power are obtained against various classes of alternative distribution functions.     

On some nonparametric tests for profile analysis of several multivariate samples

For profile analysis of independent samples from several multivariate populations, a nonparametric analog of the hypothesis of parallelism of population profiles is formulated. A class of asymptotically distribution-free statistics is offered to test this hypothesis. These are based on generalized U statistics and are in some sense modifications of statistics offered previously by one of the authors for testing the homogeneity hypothesis. Consistency of these statistics is established for suitable alternatives and also asymptotic power is investigated.     

Separation theorems for singular values of matrices and their applications in multivariate analysis

Separation theorems for singular values of a matrix, similar to the Poincaré separation theorem for the eigenvalues of a Hermitian matrix, are proved. The results are applied to problems in approximating a given r.v. by an r.v. in a specified class. In particular, problems of canonical correlations, reduced rank regression, fitting an orthogonal random variable (r.v.) to a given r.v., and estimation of residuals in the Gauss-Markoff model are discussed. In each case, a solution is obtained by minimizing a suitable norm. In some cases a common solution is shown to minimize a wide class of norms known as unitarily invariant norms introduced by von Neumann.     

On tests for selection of variables and independence under multivariate regression models

The authors consider various procedures for testing the hypotheses of independence of two sets of variables and certain regression coefficients are zero under multivariate regression model. Various properties of these procedures and the asymptotic distributions associated with these procedures are also considered.     

On the quadratic estimation of covariance matrices in multivariate linear models

This paper investigates the estimation of covariance matrices in multivariate mixed models. Some sufficient conditions are derived for a multivariate quadratic form and a linear combination of multivariate quadratic forms to be the BQUE (quadratic unbiased and severally minimum varianced) estimators of its expectations.     

Nonnull distributions of some statistics associated with testing for the equality of two covariance matrices

The nonnull distribution of some statistics, used for testing Σ₁ = Σ₂ are obtained as mixtures of incomplete beta functions as well as mixtures of incomplete gamma functions. The introduction of the convergence factors and certain recurrence relations are useful in the computation of the power of the tests as well as computation of exact percentage points for tests of significance.     

Asymptotic distributions for the elementary symmetric functions of two matrices under the assumption of linearity

The asymptotic distributions of the elementary symmetric functions (esf's) of the characteristic roots of a noncentral multivariate beta matrix and of the generalized correlation matrix (noncentral under the assumption of linearity) are derived.     

Two testing problems relating the real and complex multivariate normal distributions

The first problem considered is that of testing for the reality of the covariance matrix of a p-dimensional complex normal distribution, while the second is that of testing that a 2p-dimensional real normal distribution has a p-dimensional complex structure. Both problems are reduced by invariance to their maximal invariant statistics, and the null and non-null distributions of these are obtained. Complete classes of unbiased, invariant tests are described for both problems, the locally most powerful invariant tests are obtained, and the admissibility of the likelihood ratio tests is established.     

Computation of the null distribution of the largest or smallest latent roots of a beta matrix

An algorithm is presented for the numerical evaluation of the null distribution of the largest latent root of a beta matrix, based on a finite series recently given by Khatri [6]. The same method can also be used for the distribution of the smallest latent root, and it can be easily adapted to find percentage points. The method is only useful if the size of the matrix and the size of the denominator sample are small, and in this sense it complements some of the large sample approximations.Although the calculation may be fairly lengthy, the algorithm itself is quite short, and a Fortran coding of it will be submitted to the Journal of the Royal Statistical Society, Series C (Applied Statistics) for publication.     

An asymptotic decomposition for multivariate distribution-free tests of independence

In the multivariate case, the empirical dependence function, defined as the empirical distribution function with reduced uniform margins on the unit interval, can be shown for an i.i.d. sequence to converge weakly in an asymptotic way to a limiting Gaussian process. The main result of this paper is that this limiting process can be canonically separated into a finite set of independent Gaussian processes, enabling one to test the existence of dependence relationships within each subset of coordinates independently (in an asymptotic way) of what occurs in the other subsets. As an application we derive the Karhunen-Loeve expansions of the corresponding processes and give the limiting distribution of the multivariate Cramer-Von Mises test of independence, generalizing results of Blum, Kiefer, Rosenblatt, and Dugué. Other extensions are mentioned, including a generalization of Kendall's τ.     

On monotonicity of the modified likelihood ratio test for the equality of two covariances

For testing the hypothesis of equality of two covariances (Σ₁ and Σ₂) of two p-dimensional multivariate normal populations, it is shown that the power function of the modified likelihood ratio test increases as λ₁ increases from one and λ_r decreases from one where λ₁ > … > λ_r > 0 are the distinct characteristic roots of Σ₁Σ₂^?1, r ≤ p. As a by-product we get the unbiased result already established by Sugiura and Nagao (1968).     

On the use of the directional derivative in obtaining multivariate extreme values

Current use of the directional derivative appears, with notable exceptions such as Whittle (1971, 1973) and Vainberg (1973), to be limited largely to textbooks on advanced calculus, and to spaces of at most three dimensions. The present paper develops a calculus of the directional derivative for arbitrary finite dimensional vector spaces. Applications are made to classical maximum likelihood estimation in the case of the multivariate normal density and to other multivariate problems involving stationary points.     

Monotonicity of the power functions of modified likelihood ratio criterion for the homogeneity of variances and of the sphericity test

The modified likelihood ratio criterion for testing the homogeneity of variances of p univariate normal populations, and the sphericity test, are both shown in this paper to have a monotone nondecreasing power function.     

Strong approximations of the Hoeffding,Blum, Kiefer,Rosenblatt multivariate empirical process

Hoeffding (Ann. Math. Statist. 1948) and Blum, Kiefer and Rosenblatt (Ann. Math. Statist. 1961) constructed distribution free tests of independence based on a multivariate empirical process. We establish strong invariance principles for the latter and also for appropriate functionals of it.     

On the evaluation of some distributions that arise in simultaneous tests for the equality of the latent roots of the covariance matrix

In this paper, the authors consider the evaluation of the distribution functions of the ratios of the intermediate roots to the trace of the real Wishart matrix as well as the ratios of the individual roots to the trace of the complex Wishart matrix. In addition, the authors consider the evaluation of the distribution functions of the ratios of the extreme roots of the Wishart matrix in the real and complex cases. Some applications and tables of the above distributions are also given.     

On the performance of some statistics in discriminant analysis based on covariates

In this paper we study the behavior of three statistics suggested for testing the hypothesis, H₀ : μ₁ = μ₂, in the two sample case, in the presence of covariables. Power comparisons are made in the case when δ₂, the difference of the mean vectors in the covariates, is not equal to zero. This extends an earlier paper of the authors [Sanklya Ser. B35 51–78], where δ₂ was assumed to be equal to zero. The results reiterate those obtained in the above cited paper that for low observed values of D_q² one would use t₂ otherwise t₃ would be recommended. The statistic t₁ does not seem to be appropriate for testing this hypothesis.     

Nonparametric estimation of location parameter after a preliminary test on regression in the multivariate case

For a simple multivariate regression model, nonparametric estimation of the (vector of) intercept following a preliminary test on the regression vector is considered. Along with the asymptotic distribution of these estimators, their asymptotic bias and dispersion matrices are studied and allied efficiency results are presented.     

On the robustness of least squares procedures in regression models

The criterion robustness of the standard likelihood ratio test (LRT) under the multivariate normal regression model and also the inference robustness of the same test under the univariate set up are established for certain nonnormal distributions of errors. Restricting attention to the normal distribution of errors in the context of univariate regression models, conditions on the design matrix are established under which the usual LRT of a linear hypothesis (under homoscedasticity of errors) remains valid if the errors have an intraclass covariance structure. The conditions hold in the case of some standard designs. The relevance of C. R. Rao's (1967 In Proceedings Fifth Berkeley Symposium on Math. Stat. and Prob., Vol. 1, pp. 355–372) and G. Zyskind's (1967, Ann. Math. Statist.38 1092–1110) conditions in this context is discussed.