Near-exact distributions for the independence and sphericity likelihood ratio test statistics

In this paper we show how, based on a decomposition of the likelihood ratio test for sphericity into two independent tests and a suitably developed decomposition of the characteristic function of the logarithm of the likelihood ratio test statistic to test independence in a set of variates, we may obtain extremely well-fitting near-exact distributions for both test statistics. Since both test statistics have the distribution of the product of independent Beta random variables, it is possible to obtain near-exact distributions for both statistics in the form of Generalized Near-Integer Gamma distributions or mixtures of these distributions. For the independence test statistic, numerical studies and comparisons with asymptotic distributions proposed by other authors show the extremely high accuracy of the near-exact distributions developed as approximations to the exact distribution. Concerning the sphericity test statistic, comparisons with formerly developed near-exact distributions show the advantages of these new near-exact distributions.     

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).     

The likelihood ratio test for homogeneity in bivariate normal mixtures

This paper investigates the asymptotic properties of the likelihood ratio statistic for testing homogeneity in a bivariate normal mixture model with known covariance. The asymptotic null distributions of the likelihood ratio statistic and a modified likelihood ratio statistic are obtained in explicit form. The distributions are identical. The results of a small simulation study to approximate the null distribution are presented.     

The distribution of the sphericity test criterion

The exact distribution of Mauchly's sphericity test criterion W = |S|/[trS/p]^p, when S is the sum of product matrix from a sample of size N taken from a p-variate normal population, is obtained using contour integration and methods similar to those of Nair and Box. Tables of percentage points for p = 4(1)10, α = 0.01 and 0.05, and various values of N (including small) are given and comparisons made with approximate percentage points using methods of Box, Mauchly, Tukey and Wilks, and Davis.     

A likelihood ratio test for separability of covariances

We propose a formal test of separability of covariance models based on a likelihood ratio statistic. The test is developed in the context of multivariate repeated measures (for example, several variables measured at multiple times on many subjects), but can also apply to a replicated spatio-temporal process and to problems in meteorology, where horizontal and vertical covariances are often assumed to be separable. Separable models are a common way to model spatio-temporal covariances because of the computational benefits resulting from the joint space-time covariance being factored into the product of a covariance function that depends only on space and a covariance function that depends only on time. We show that when the null hypothesis of separability holds, the distribution of the test statistic does not depend on the type of separable model. Thus, it is possible to develop reference distributions of the test statistic under the null hypothesis. These distributions are used to evaluate the power of the test for certain nonseparable models. The test does not require second-order stationarity, isotropy, or specification of a covariance model. We apply the test to a multivariate repeated measures problem.     

The power of the likelihood ratio test for additional information in a multivariate linear model

Summary This paper deals with the likelihood ratio test for additional information in a multivariate linear model. It is shown that the power of the likelihood ratio test procedure has a monotonicity property. Asymptotic approximations for the power are also obtained.     

Nonnull distribution of likelihood ratio criterion for reality of covariance matrix

In this paper the distribution of the likelihood ratio test for testing the reality of the covariance matrix of a complex multivariate normal distribution is investigated. Some simplifications in the noncentral distribution are made and the noncentral distribution is derived for the special case where the rank of the noncentrality matrix is two. In the null case exact expressions for the distribution are given up to p = 6, and percentage points are tabulated. These percentage points were compared with percentage points derived from an asymptotic expansion of the distribution, and the accuracy of the approximation was found to be sufficient for several practical situations.     

A new test for sphericity of the covariance matrix for high dimensional data

In this paper we propose a new test procedure for sphericity of the covariance matrix when the dimensionality, p, exceeds that of the sample size, N=n+1. Under the assumptions that (A) as p→∞ for i=1,…,16 and (B) p/n→c<∞ known as the concentration, a new statistic is developed utilizing the ratio of the fourth and second arithmetic means of the eigenvalues of the sample covariance matrix. The newly defined test has many desirable general asymptotic properties, such as normality and consistency when (n,p)→∞. Our simulation results show that the new test is comparable to, and in some cases more powerful than, the tests for sphericity in the current literature.     

The block sphericity test—exact and near‐exact distributions for the likelihood ratio statistic

Using a suitable decomposition of the null hypothesis of the sphericity test for several blocks of variables, into a sequence of conditionally independent null hypotheses, we show that it is possible to obtain the expressions for the likelihood ratio test statistic, for its hth null moment, and for the characteristic function of its logarithm. The exact distribution of the logarithm of the likelihood ratio test statistic is obtained in the form of a sum of a generalized integer gamma distribution with the sum of a given number of independent logbeta distributions, taking the form of a single generalized integer gamma distribution when each set of variables has two variables. The development of near‐exact distributions arises, from the previous decomposition of the null hypothesis and from the consequent‐induced factorization of the characteristic function, as a natural and practical way to approximate the exact distribution of the test statistic. A measure based on the exact and approximating characteristic functions, which gives an upper bound on the distance between the corresponding distribution functions, is used to assess the quality of the near‐exact distributions proposed and to compare them with an asymptotic approximation on the basis of Box's method. Copyright © 2011 John Wiley & Sons, Ltd.     

The likelihood ratio tests for the dimensionality of regression coefficients

In this paper we give a unified derivation of the likelihood ratio (LR) statistics for testing the hypothesis on the dimensionality of regression coefficients under a usual MANOVA model. We also derive the LR statistics under a general MANOVA model and study their asymptotic null and nonnull distributions. Further it is shown that the test statistic used by Bartlett [4] for testing the hypothesis that the last p?k canonical correlations are all zero is the LR statistic.     

Some optimization problems with applications to canonical correlations and sphericity tests

Optimization problems are connected with maximization of three functions, namely, geometric mean, arithmetic mean and harmonic mean of the eigenvalues of (X′ΣX)^?1ΣY(Y′ΣY)^?1Y′ΣX, where Σ is positive definite, X and Y are p × r and p × s matrices of ranks r and s (≥r), respectively, and X′Y = 0. Some interpretations of these functions are given. It is shown that the maximum values of these functions are obtained at the same point given by X = (h₁ + ?₁h_p, …, h_r + ?_rh_p?r+1) and $\text{Y = (h}{}_1\text{? ?}{}_1\text{h}{}_{\mathrm{p}}\text{, …, h}{}_{\mathrm{r}}\text{? ?}{}_{\mathrm{r}}\text{h}{}_{\mathrm{p?r+1}}\text{,}\text{Y}{}_{\mathrm{r+1}}\text{, …,}\text{Y}{}_{\mathrm{s}}\text{)}$, where h₁, …, h_p are the eigenvectors of Σ corresponding to the eigenvalues λ₁ ≥ λ₂ ≥ … ≥ λ_p > 0, ?_j = +1 or ?1 for j = 1,2,…, r and $\text{Y}{}_{\mathrm{r+1}}\text{, …,}\text{Y}{}_{\mathrm{s}}$, are linear functions of h_r+1,…, h_p?r. These results are extended to intermediate stationary values. They are utilized in obtaining the inequalities for canonical correlations θ₁,…,θ_r and they are given by expressions (3.8)–(3.10). Further, some new union-intersection test procedures for testing the sphericity hypothesis are given through test statistics (3.11)–(3.13).     

Some implications of the union-intersection principle for tests of sphericity

The sphericity hypothesis may be expressed as an intersection of simpler hypotheses on the invariant subspaces of the variance matrix. Applying the union-intersection principle to dissections of this type establishes a link between tests of independence and tests of sphericity. We use some recent results of Bloomfield and Watson [2] and Knott [4] to derive a class of union-intersection tests for sphericity from likelihood ratio tests of independence of two sets of variates. As well, we show that the ordinary likelihood ratio test for sphericity has a natural union-intersection interpretation.     

UMVU estimation of the ratio of powers of normal generalized variances under correlation

We consider estimation of the ratio of arbitrary powers of two normal generalized variances based on two correlated random samples. First, the result of Iliopoulos [Decision theoretic estimation of the ratio of variances in a bivariate normal distribution, Ann. Inst. Statist. Math. 53 (2001) 436-446] on UMVU estimation of the ratio of variances in a bivariate normal distribution is extended to the case of the ratio of any powers of the two variances. Motivated by these estimators’ forms we derive the UMVU estimator in the multivariate case. We show that it is proportional to the ratio of the corresponding powers of the two sample generalized variances multiplied by a function of the sample canonical correlations. The mean squared errors of the derived UMVU estimator and the maximum likelihood estimator are compared via simulation for some special cases.     

Modified likelihood ratio test for homogeneity in normal mixtures with two samples

This paper investigates the modified likelihood ratio test（LRT） for homogeneity in normal mixtures of two samples with mixing proportions unknown. It is proved that the limit distribution of the modified likelihood ratio test is X^2（1）.     

The covariance structure and likelihood function for multivariate dyadic data

This paper extends the results in Li and Loken [A unified theory of statistical analysis and inference for variance component models for dyadic data, Statist. Sinica 12 (2002) 519-535] on the statistical analysis of measurements taken on dyads to the situations in which more than one attribute are measured on each dyad. Starting from the covariance structure for the univariate case obtained in Li and Loken (2002), the covariance structure for the multivariate case is derived based on the group symmetry induced by the assumed exchangeability in the units. Our primary objective is to document the Gaussian likelihood and the sufficient statistics for multivariate dyadic data in closed form, so that they can be referenced by researchers as they analyze those data. The derivation carried out can also serve as an example of multivariate extension of univariate models based on exchangeability.     

Distribution of the likelihood ratio criterion for testing Σ = Σ0, μ = μ0

The exact null distribution of the likelihood ratio criterion for testing H₀: Σ = Σ₀ and μ = μ₀ against alternatives H₁: Σ ≠ Σ₀ or μ ≠ μ₀ in N_p(μ, Σ) has been obtained as (a) a chi-square series and (b) a beta series. Percentage points have been tabulated for p = 2(1) 6, α = .005, .01, .025, .05, .1, and .25 and various values of sample size N.     

On the comparison of the pre-test and shrinkage phi-divergence test estimators for the symmetry model of categorical data

The estimation problem of the parameters in a symmetry model for categorical data has been considered for many authors in the statistical literature (for example, Bowker (1948) [1], Ireland et al. (1969) [2], Quade and Salama (1975) [3], Cressie and Read (1988) [4], Menéndez et al. (2005) [5]) without using uncertain prior information. It is well known that many new and interesting estimators, using uncertain prior information, have been studied by a host of researchers in different statistical models, and many papers have been published on this topic (see Saleh (2006) [9] and references therein). In this paper, we consider the symmetry model of categorical data and we study, for the first time, some new estimators when non-sample information about the symmetry of the probabilities is considered. The decision to use a “restricted” estimator or an “unrestricted” estimator is based on the outcome of a preliminary test, and then a shrinkage technique is used. It is interesting to note that we present a unified study in the sense that we consider not only the maximum likelihood estimator and likelihood ratio test or chi-square test statistic but we consider minimum phi-divergence estimators and phi-divergence test statistics. Families of minimum phi-divergence estimators and phi-divergence test statistics are wide classes of estimators and test statistics that contain as a particular case the maximum likelihood estimator, likelihood ratio test and chi-square test statistic. In an asymptotic set-up, the biases and the risk under the squared loss function for the proposed estimators are derived and compared. A numerical example clarifies the content of the paper.     

Exact nonnull distribution of Wilks’ statistic: The ratio and product of independent components

The study of the noncentral matrix variate beta type distributions has been sidelined because the final expressions for the densities depend on an integral that has not been resolved in an explicit way. We derive an exact expression for the nonnull distribution of Wilks’ statistic and precise expressions for the densities of the ratio and product of two independent components of matrix variates where one matrix variate has the noncentral matrix variate beta type I distribution and the other has the matrix variate beta type I distribution. We provide the expressions for the densities of the determinant of the ratio and the product of these two components. These distributions play a fundamental role in various areas of statistics, for example in the criteria proposed by Wilks.     

A high-dimensional test for the equality of the smallest eigenvalues of a covariance matrix

For the test of sphericity, Ledoit and Wolf [Ann. Statist. 30 (2002) 1081-1102] proposed a statistic which is robust against high dimensionality. In this paper, we consider a natural generalization of their statistic for the test that the smallest eigenvalues of a covariance matrix are equal. Some inequalities are obtained for sums of eigenvalues and sums of squared eigenvalues. These bounds permit us to obtain the asymptotic null distribution of our statistic, as the dimensionality and sample size go to infinity together, by using distributional results obtained by Ledoit and Wolf [Ann. Statist. 30 (2002) 1081-1102]. Some empirical results comparing our test with the likelihood ratio test are also given.