Some tests for the covariance matrix with fewer observations than the dimension under non-normality

This article analyzes whether some existing tests for the p×p covariance matrix Σ of the N independent identically distributed observation vectors work under non-normality. We focus on three hypotheses testing problems: (1) testing for sphericity, that is, the covariance matrix Σ is proportional to an identity matrix I_p; (2) the covariance matrix Σ is an identity matrix I_p; and (3) the covariance matrix is a diagonal matrix. It is shown that the tests proposed by Srivastava (2005) for the above three problems are robust under the non-normality assumption made in this article irrespective of whether N≤p or N≥p, but (N,p)→∞, and N/p may go to zero or infinity. Results are asymptotic and it may be noted that they may not hold for finite (N,p).     

Characterizations of Arnold and Strauss’ and related bivariate exponential models

Characterizations of probability distributions is a topic of great popularity in applied probability and reliability literature for over last 30 years. Beside the intrinsic mathematical interest (often related to functional equations) the results in this area are helpful for probabilistic and statistical modelling, especially in engineering and biostatistical problems. A substantial number of characterizations has been devoted to a legion of variants of exponential distributions. The main reliability measures associated with a random vector X are the conditional moment function defined by m_φ(x)=E(φ(X)|X?x) (which is equivalent to the mean residual life function e(x)=m_φ(x)-x when φ(x)=x) and the hazard gradient function h(x)=-∇logR(x), where R(x) is the reliability (survival) function, R(x)=Pr(X?x), and ∇ is the operator . In this paper we study the consequences of a linear relationship between the hazard gradient and the conditional moment functions for continuous bivariate and multivariate distributions. We obtain a general characterization result which is the applied to characterize Arnold and Strauss’ bivariate exponential distribution and some related models.     

The restricted EM algorithm under inequality restrictions on the parameters

One of the most powerful algorithms for maximum likelihood estimation for many incomplete-data problems is the EM algorithm. The restricted EM algorithm for maximum likelihood estimation under linear restrictions on the parameters has been handled by Kim and Taylor (J. Amer. Statist. Assoc. 430 (1995) 708-716). This paper proposes an EM algorithm for maximum likelihood estimation under inequality restrictions A₀β?0, where β is the parameter vector in a linear model W=Xβ+ε and ε is an error variable distributed normally with mean zero and a known or unknown variance matrix Σ>0. Some convergence properties of the EM sequence are discussed. Furthermore, we consider the consistency of the restricted EM estimator and a related testing problem.     

Kernel estimation of density level sets

Let f be a multivariate density and f_n be a kernel estimate of f drawn from the n-sample X₁,…,X_n of i.i.d. random variables with density f. We compute the asymptotic rate of convergence towards 0 of the volume of the symmetric difference between the t-level set {f?t} and its plug-in estimator {f_n?t}. As a corollary, we obtain the exact rate of convergence of a plug-in-type estimate of the density level set corresponding to a fixed probability for the law induced by f.     

On the dependence between the extreme order statistics in the proportional hazards model

Let X₁,…,X_n be a random sample from an absolutely continuous distribution with non-negative support, and let Y₁,…,Y_n be mutually independent lifetimes with proportional hazard rates. Let also X₍₁₎<?<X_(n) and Y₍₁₎<?<Y_(n) be their associated order statistics. It is shown that the pair (X₍₁₎,X_(n)) is then more dependent than the pair (Y₍₁₎,Y_(n)), in the sense of the right-tail increasing ordering of Avérous and Dortet-Bernadet [LTD and RTI dependence orderings, Canad. J. Statist. 28 (2000) 151-157]. Elementary consequences of this fact are highlighted.     

Characterization of Wishart-Laplace distributions via Jordan algebra homomorphisms

For a real, Hermitian, or quaternion normal random matrix Y with mean zero, necessary and sufficient conditions for a quadratic form Q(Y) to have a Wishart-Laplace distribution (the distribution of the difference of two independent central Wishart W_p(m_i,Σ) random matrices) are given in terms of a certain Jordan algebra homomorphism ρ. Further, it is shown that {Q_k(Y)} is independent Laplace-Wishart if and only if in addition to the aforementioned conditions, the images ρ_k(Σ⁺) of the Moore-Penrose inverse Σ⁺ of Σ are mutually orthogonal: ρ_k(Σ⁺)ρ_?(Σ⁺)=0 for k≠?.     

More on the inadmissibility of step-up

Cohen and Sackrowitz [Characterization of Bayes procedures for multiple endpoint problems and inadmissibility of the step-up procedure, Ann. Statist. 33 (2005) 145-158] proved that the step-up multiple testing procedure is inadmissible for a multivariate normal model with unknown mean vector and known intraclass covariance matrix. The hypotheses tested are each mean is zero vs. each mean is positive. The risk function is a 2×1 vector where one component is average size and the other component is one minus average power. In this paper, we extend the inadmissibility result to several different models, to two-sided alternatives, and to other risk functions. The models include one-parameter exponential families, independent t-variables, independent χ²-variables, t-tests arising from the analysis of variance, and t-tests arising from testing treatments against a control. The additional risk functions are linear combinations where one component is the false discovery rate (FDR).     

The use of the tetrachoric series for evaluating multivariate normal probabilities

The tetrachoric series is a technique for evaluating multivariate normal probabilities frequently cited in the statistical literature. In this paper we have examined the convergence properties of the tetrachoric series and have established the following. For orthant probabilities, the tetrachoric series converges if $\text{|;?}{}_{\mathrm{ij}}\text{|; <}\text{1}\text{(k ? 1)}$, 1 ≤ i < j ≤ k, where ?_ij are the correlation coefficients of a k-variate normal distribution. The tetrachoric series for orthant probabilities diverges whenever k is even and $\text{?}{}_{\mathrm{ij}}\text{>}\text{1}\text{(k ? 1)}$ or k is odd and $\text{?}{}_{\mathrm{ij}}\text{>}\text{1}\text{(k ? 2)}$, 1 ≤ i < j ≤ k. Other specific results concerning the convergence or divergence of this series are also given. The principal point is that the assertion that the tetrachoric series converges for all k ≥ 2 and all ?_ij such that the correlation matrix is positive definite is false.     

Wishart ratios with dependent structure: New members of the bimatrix beta type IV

In multivariate statistics under normality, the problems of interest are random covariance matrices (known as Wishart matrices) and “ratios” of Wishart matrices that arise in multivariate analysis of variance (MANOVA) (see 24). The bimatrix variate beta type IV distribution (also known in the literature as bimatrix variate generalised beta; matrix variate generalization of a bivariate beta type I) arises from “ratios” of Wishart matrices. In this paper, we add a further independent Wishart random variate to the “denominator” of one of the ratios; this results in deriving the exact expression for the density function of the bimatrix variate extended beta type IV distribution. The latter leads to the proposal of the bimatrix variate extended F distribution. Some interesting characteristics of these newly introduced bimatrix distributions are explored. Lastly, we focus on the bivariate extended beta type IV distribution (that is an extension of bivariate Jones’ beta) with emphasis on P(X₁<X₂) where X₁ is the random stress variate and X₂ is the random strength variate.     

Maximum entropy characterizations of the multivariate Liouville distributions

A random vector X=(X₁,X₂,…,X_n) with positive components has a Liouville distribution with parameter θ=(θ₁,θ₂,…,θ_n) if its joint probability density function is proportional to , θ_i>0 [R.D. Gupta, D.S.P. Richards, Multivariate Liouville distributions, J. Multivariate Anal. 23 (1987) 233-256]. Examples include correlated gamma variables, Dirichlet and inverted Dirichlet distributions. We derive appropriate constraints which establish the maximum entropy characterization of the Liouville distributions among all multivariate distributions. Matrix analogs of the Liouville distributions are considered. Some interesting results related to I-projection from a Liouville distribution are presented.     

A statistical model for random rotations

This paper studies the properties of the Cayley distributions, a new family of models for random p×p rotations. This class of distributions is related to the Cayley transform that maps a p(p-1)/2×1 vector s into SO(p), the space of p×p rotation matrices. First an expression for the uniform measure on SO(p) is derived using the Cayley transform, then the Cayley density for random rotations is investigated. A closed-form expression is derived for its normalizing constant, a simple simulation algorithm is proposed, and moments are derived. The efficiencies of moment estimators of the parameters of the new model are also calculated. A Monte Carlo investigation of tests and of confidence regions for the parameters of the new density is briefly summarized. A numerical example is presented.     

Estimation of regression contour clusters—an application of the excess mass approach to regression

The paper shows that the technique known as excess mass can be translated to non-parametric regression with random design in d-dimensional Euclidean space, where the regression function m is given by m(x)=E(Y∣X=x),x∈R^d. The approach is applied to estimating regression contour clusters, which are sets where m exceeds a certain threshold value. This is accomplished without prior estimation of the regression function. Consistency of the resulting estimators is studied, and a functional central limit theorem for the excess mass is derived in the regression context.     

Asymptotic distributions of robust shape matrices and scales

It has been frequently observed in the literature that many multivariate statistical methods require the covariance or dispersion matrix Σ of an elliptical distribution only up to some scaling constant. If the topic of interest is not the scale but only the shape of the elliptical distribution, it is not meaningful to focus on the asymptotic distribution of an estimator for Σ or another matrix Γ∝Σ. In the present work, robust estimators for the shape matrix and the associated scale are investigated. Explicit expressions for their joint asymptotic distributions are derived. It turns out that if the joint asymptotic distribution is normal, the estimators presented are asymptotically independent for one and only one specific choice of the scale function. If it is non-normal (this holds for example if the estimators for the shape matrix and scale are based on the minimum volume ellipsoid estimator) only the scale function presented leads to asymptotically uncorrelated estimators. This is a generalization of a result obtained by Paindaveine [D. Paindaveine, A canonical definition of shape, Statistics and Probability Letters 78 (2008) 2240-2247] in the context of local asymptotic normality theory.     

Latent models for cross-covariance

We consider models for the covariance between two blocks of variables. Such models are often used in situations where latent variables are believed to present. In this paper we characterize exactly the set of distributions given by a class of models with one-dimensional latent variables. These models relate two blocks of observed variables, modeling only the cross-covariance matrix. We describe the relation of this model to the singular value decomposition of the cross-covariance matrix. We show that, although the model is underidentified, useful information may be extracted. We further consider an alternative parameterization in which one latent variable is associated with each block, and we extend the result to models with r-dimensional latent variables.     

Tests on multiple correlation coefficient and multiple partial correlation coefficient

Let X₁(1 × 1), X₂(1 × p₂), and X₃(1 × p₃) be three sets of random variables distributed jointly as a normal distribution. Let ?_1.23 and ?_1.2 be the multiple correlation coefficients between X₁ and (X₂, X₃) and between X₁ and X₂, respectively. Invariant tests for the following four testing problems are considered: ?_1.23 = 0 vs ?_1.23 > 0, ?_1.23 = 0, vs ?_1.23 > 0, ?_1.23 = ?_1.2 vs ?_1.23 > ?_1.2, ?_1.2 = 0 vs ?_1.2 > 0. The joint distribution of the sample multiple correlation coefficients R_1.23 and R_1.2 is derived.     

Estimation of the mean vector of a multivariate normal distribution: subspace hypothesis

This paper considers the estimation of the mean vector θ of a p-variate normal distribution with unknown covariance matrix Σ when it is suspected that for a p×r known matrix B the hypothesis θ=Bη, η∈R^r may hold. We consider empirical Bayes estimators which includes (i) the unrestricted unbiased (UE) estimator, namely, the sample mean vector (ii) the restricted estimator (RE) which is obtained when the hypothesis θ=Bη holds (iii) the preliminary test estimator (PTE), (iv) the James-Stein estimator (JSE), and (v) the positive-rule Stein estimator (PRSE). The biases and the risks under the squared loss function are evaluated for all the five estimators and compared. The numerical computations show that PRSE is the best among all the five estimators even when the hypothesis θ=Bη is true.     

New families of estimators and test statistics in log-linear models

In this paper we consider categorical data that are distributed according to a multinomial, product-multinomial or Poisson distribution whose expected values follow a log-linear model and we study the inference problem of hypothesis testing in a log-linear model setting. The family of test statistics considered is based on the family of ?-divergence measures. The unknown parameters in the log-linear model under consideration are also estimated using ?-divergence measures: Minimum ?-divergence estimators. A simulation study is included to find test statistics that offer an attractive alternative to the Pearson chi-square and likelihood-ratio test statistics.     

Uniformly minimum variance nonnegative quadratic unbiased estimation in a generalized growth curve model

Consider the generalized growth curve model subject to R(X_m)⊆?⊆R(X₁), where B_i are the matrices of unknown regression coefficients, and E=(ε₁,…,ε_s)^′ and are independent and identically distributed with the same first four moments as a random vector normally distributed with mean zero and covariance matrix Σ. We derive the necessary and sufficient conditions under which the uniformly minimum variance nonnegative quadratic unbiased estimator (UMVNNQUE) of the parametric function with C≥0 exists. The necessary and sufficient conditions for a nonnegative quadratic unbiased estimator with of to be the UMVNNQUE are obtained as well.     

On Pickands coordinates in arbitrary dimensions

Pickands coordinates were introduced as a crucial tool for the investigation of bivariate extreme value models. We extend their definition to arbitrary dimensions and, thus, we can generalize many known results for bivariate extreme value and generalized Pareto models to higher dimensions and arbitrary extreme value margins.In particular we characterize multivariate generalized Pareto distributions (GPs) and spectral δ-neighborhoods of GPs in terms of best attainable rates of convergence of extremes, which are well-known results in the univariate case. A sufficient univariate condition for a multivariate distribution function (df) to belong to the domain of attraction of an extreme value df is derived. Bounds for the variational distance in peaks-over-threshold models are established, which are based on Pickands coordinates.     

Unbiased invariant minimum norm estimation in generalized growth curve model

This paper considers the generalized growth curve model subject to R(X_m)⊆R(X_m-1)⊆?⊆R(X₁), where B_i are the matrices of unknown regression coefficients, X_i,Z_i and U are known covariate matrices, i=1,2,…,m, and E splits into a number of independently and identically distributed subvectors with mean zero and unknown covariance matrix Σ. An unbiased invariant minimum norm quadratic estimator (MINQE(U,I)) of tr(CΣ) is derived and the conditions for its optimality under the minimum variance criterion are investigated. The necessary and sufficient conditions for MINQE(U,I) of tr(CΣ) to be a uniformly minimum variance invariant quadratic unbiased estimator (UMVIQUE) are obtained. An unbiased invariant minimum norm quadratic plus linear estimator (MINQLE(U,I)) of is also given. To compare with the existing maximum likelihood estimator (MLE) of tr(CΣ), we conduct some simulation studies which show that our proposed estimator performs very well.