Canonical representation of conditionally specified multivariate discrete distributions

Most work on conditionally specified distributions has focused on approaches that operate on the probability space, and the constraints on the probability space often make the study of their properties challenging. We propose decomposing both the joint and conditional discrete distributions into characterizing sets of canonical interactions, and we prove that certain interactions of a joint distribution are shared with its conditional distributions. This invariance opens the door for checking the compatibility between conditional distributions involving the same set of variables. We formulate necessary and sufficient conditions for the existence and uniqueness of discrete conditional models, and we show how a joint distribution can be easily computed from the pool of interactions collected from the conditional distributions. Hence, the methods can be used to calculate the exact distribution of a Gibbs sampler. Furthermore, issues such as how near compatibility can be reconciled are also discussed. Using mixed parametrization, we show that the proposed approach is based on the canonical parameters, while the conventional approaches are based on the mean parameters. Our advantage is partly due to the invariance that holds only for the canonical parameters.     

Estimation of means of multivariate normal populations with order restriction

Multivariate isotonic regression theory plays a key role in the field of statistical inference under order restriction for vector valued parameters. Two cases of estimating multivariate normal means under order restricted set are considered. One case is that covariance matrices are known, the other one is that covariance matrices are unknown but are restricted by partial order. This paper shows that when covariance matrices are known, the estimator given by this paper always dominates unrestricted maximum likelihood estimator uniformly, and when covariance matrices are unknown, the plug-in estimator dominates unrestricted maximum likelihood estimator under the order restricted set of covariance matrices. The isotonic regression estimators in this paper are the generalizations of plug-in estimators in unitary case.     

Point prediction of future order statistics from an exponential distribution

Two-sample point prediction is considered for a two-parameter exponential distribution. Several point predictors such as the best unbiased predictor, best invariant predictor and maximum likelihood predictor are obtained for future order statistics on the basis of observed record values in two cases: where the location parameter is known and unknown. These predictors are compared in the sense of their mean squared prediction errors. Finally, some numerical results are given to illustrate the proposed procedures.     

Distribution of quadratic forms under skew normal settings

For a class of multivariate skew normal distributions, the noncentral skew chi-square distribution is studied. The necessary and sufficient conditions under which a sequence of quadratic forms is generalized noncentral skew chi-square distributed random variables are obtained. Several examples are given to illustrate the results.     

Progressive Type II censored order statistics for multivariate observations

For a sequence of independent and identically distributed random vectors , i=1,2,…,n, we consider the conditional ordering of these random vectors with respect to the magnitudes of , where N is a p-variate continuous function defined on the support set of X₁ and satisfying certain regularity conditions. We also consider the Progressive Type II right censoring for multivariate observations using conditional ordering. The need for the conditional ordering of random vectors exists for example, in reliability analysis when a system has n independent components each consisting of p arbitrarily dependent and parallel connected elements. Let the vector of life lengths for the ith component of the system be , where denotes the life length of the jth element of the ith component. Then the first failure in the system occurs at time , and for this case . In this paper we introduce the conditionally ordered and Progressive Type II right-censored conditionally ordered statistics for multivariate observations and to study their distributional properties.     

A weighted multivariate signed-rank test for cluster-correlated data

A weighted multivariate signed-rank test is introduced for an analysis of multivariate clustered data. Observations in different clusters may then get different weights. The test provides a robust and efficient alternative to normal theory based methods. Asymptotic theory is developed to find the approximate p-value as well as to calculate the limiting Pitman efficiency of the test. A conditionally distribution-free version of the test is also discussed. The finite-sample behavior of different versions of the test statistic is explored by simulations and the new test is compared to the unweighted and weighted versions of Hotelling’s T² test and the multivariate spatial sign test introduced in [D. Larocque, J. Nevalainen, H. Oja, A weighted multivariate sign test for cluster-correlated data, Biometrika 94 (2007) 267-283]. Finally, a real data example is used to illustrate the theory.     

Asymptotic power comparison of three tests in GMANOVA when the number of observed points is large

This paper is concerned with the testing problem of generalized multivariate linear hypothesis for the mean in the growth curve model(GMANOVA). Our interest is the case in which the number of the observed points p is relatively large compared to the sample size N. Asymptotic expansions of the non-null distributions of the likelihood ratio criterion, Lawley-Hotelling’s trace criterion and Bartlett-Nanda-Pillai’s trace criterion are derived under the asymptotic framework that N and p go to infinity together, while p/N→c∈(0,1). It also can be confirmed that Rothenberg’s condition on the magnitude of the asymptotic powers of the three tests is valid when p is relatively large, theoretically and numerically.     

Estimation of the precision matrix of a singular Wishart distribution and its application in high-dimensional data

In this article, the Stein-Haff identity is established for a singular Wishart distribution with a positive definite mean matrix but with the dimension larger than the degrees of freedom. This identity is then used to obtain estimators of the precision matrix improving on the estimator based on the Moore-Penrose inverse of the Wishart matrix under the Efron-Morris loss function and its variants. Ridge-type empirical Bayes estimators of the precision matrix are also given and their dominance properties over the usual one are shown using this identity. Finally, these precision estimators are used in a quadratic discriminant rule, and it is shown through simulation that discriminant methods based on the ridge-type empirical Bayes estimators provide higher correct classification rates.     

Third-order power comparisons for a class of tests for multivariate linear hypothesis under general distributions

The purpose of this paper is, in multivariate linear regression model (Part I) and GMANOVA model (Part II), to investigate the effect of nonnormality upon the nonnull distributions of some multivariate test statistics under normality. It is shown that whatever the underlying distributions, the difference of local powers up to order N⁻¹ after either Bartlett’s type adjustment or Cornish-Fisher’s type size adjustment under nonnormality coincides with that in Anderson [An Introduction to Multivariate Statistical Analysis, 2nd ed. and 3rd ed., Wiley, New York, 1984, 2003] under normality. The derivation of asymptotic expansions is based on the differential operator associated with the multivariate linear regression model under general distributions. The performance of higher-order results in finite samples, including monotone Bartlett’s type adjustment and monotone Cornish-Fisher’s type size adjustment, is examined using simulation studies.     

Consistency of the generalized MLE of a joint distribution function with multivariate interval-censored data

Wong and Yu [Generalized MLE of a joint distribution function with multivariate interval-censored data, J. Multivariate Anal. 69 (1999) 155-166] discussed generalized maximum likelihood estimation of the joint distribution function of a multivariate random vector whose coordinates are subject to interval censoring. They established uniform consistency of the generalized MLE (GMLE) of the distribution function under the assumption that the random vector is independent of the censoring vector and that both of the vector distributions are discrete. We relax these assumptions and establish consistency results of the GMLE under a multivariate mixed case interval censorship model. van der Vaart and Wellner [Preservation theorems for Glivenko-Cantelli and uniform Glivenko-Cantelli class, in: E. Gine, D.M. Mason, J.A. Wellner (Eds.), High Dimensional Probability, vol. II, Birkhäuser, Boston, 2000, pp. 115-133] and Yu [Consistency of the generalized MLE with multivariate mixed case interval-censored data, Ph.D Dissertation, Binghamton University, 2000] independently proved strong consistency of the GMLE in the L₁(μ)-topology, where μ is a measure derived from the joint distribution of the censoring variables. We establish strong consistency of the GMLE in the topologies of weak convergence and pointwise convergence, and eventually uniform convergence under appropriate distributional assumptions and regularity conditions.     

A multivariate extension of Sarhan and Balakrishnan’s bivariate distribution and its ageing and dependence properties

A new class of bivariate distributions (NBD) was recently introduced by Sarhan and Balakrishnan [A.M. Sarhan, N. Balakrishnan, A new class of bivariate distributions and its mixture, J. Multivariate Anal. 98 (2007) 1508-1527]. In this note, we give the joint survival function of a multivariate extension of the NBD, which is not an absolutely continuous multivariate distribution, and its marginal and extreme order statistics distributions are also derived. The multivariate ageing and dependence properties of the proposed n-dimensional distribution are also discussed, and then we analyze the stochastic ageing of its marginals and its minimum and maximum order statistics.     

Nonparametric likelihood based estimation for a multivariate Lipschitz density

We consider a problem of nonparametric density estimation under shape restrictions. We deal with the case where the density belongs to a class of Lipschitz functions. Devroye [L. Devroye, A Course in Density Estimation, in: Progress in Probability and Statistics, vol. 14, Birkhäuser Boston Inc., Boston, MA, 1987] considered these classes of estimates as tailor-made estimates, in contrast in some way to universally consistent estimates. In our framework we get the existence and uniqueness of the maximum likelihood estimate as well as strong consistency. This NPMLE can be easily characterized but it is not easy to compute. Some simpler approximations are also considered.     

Laplace approximations to hypergeometric functions of two matrix arguments

We present a unified approach to Laplace approximation of hypergeometric functions with two matrix arguments. The general form of the approximation is designed to exploit the Laplace approximations to hypergeometric functions of a single matrix argument presented in Butler and Wood (Ann. Statist. 30 (2002) 1155, Laplace approximations to Bessel functions of matrix argument, J. Comput. Appl. Math. 155 (2003) 359) which have proved to be very accurate in a variety of settings. All but one of the approximations presented here appear to be new. Numerical accuracy is investigated in a number of statistical applications.     

Moment properties of multivariate infinitely divisible laws and criteria for multivariate self-decomposability

Ramachandran (1969) [9, Theorem 8] has shown that for any univariate infinitely divisible distribution and any positive real number α, an absolute moment of order α relative to the distribution exists (as a finite number) if and only if this is so for a certain truncated version of the corresponding Lévy measure. A generalized version of this result in the case of multivariate infinitely divisible distributions, involving the concept of g-moments, was given by Sato (1999) [6, Theorem 25.3]. We extend Ramachandran’s theorem to the multivariate case, keeping in mind the immediate requirements under appropriate assumptions of cumulant studies of the distributions referred to; the format of Sato’s theorem just referred to obviously varies from ours and seems to have a different agenda. Also, appealing to a further criterion based on the Lévy measure, we identify in a certain class of multivariate infinitely divisible distributions the distributions that are self-decomposable; this throws new light on structural aspects of certain multivariate distributions such as the multivariate generalized hyperbolic distributions studied by Barndorff-Nielsen (1977) [12] and others. Various points relevant to the study are also addressed through specific examples.     

On Stein’s lemma, dependent covariates and functional monotonicity in multi-dimensional modeling

Tracking the correct directions of monotonicity in multi-dimensional modeling plays an important role in interpreting functional associations. In the presence of multiple predictors, we provide empirical evidence that the observed monotone directions via parametric, nonparametric or semiparametric fit of commonly used multi-dimensional models may entirely violate the actual directions of monotonicity. This breakdown is caused primarily by the dependence structure of covariates, with negligible influence from the bias of function estimation. To examine the linkage between the dependent covariates and monotone directions, we first generalize Stein’s Lemma for random variables which are mutually independent Gaussian to two important cases: dependent Gaussian, and independent non-Gaussian. We show that in both two cases, there is an explicit one-to-one correspondence between the monotone directions of a multi-dimensional function and the signs of a deterministic surrogate vector. Moreover, we demonstrate that the second case can be extended to accommodate a class of dependent covariates. This generalization further enables us to develop a de-correlation transform for arbitrarily dependent covariates. The transformed covariates preserve modeling interpretability with little loss in modeling efficiency. The simplicity and effectiveness of the proposed method are illustrated via simulation studies and real data application.     

The Stein phenomenon for monotone incomplete multivariate normal data

We establish the Stein phenomenon in the context of two-step, monotone incomplete data drawn from , a (p+q)-dimensional multivariate normal population with mean and covariance matrix . On the basis of data consisting of n observations on all p+q characteristics and an additional N−n observations on the last q characteristics, where all observations are mutually independent, denote by the maximum likelihood estimator of . We establish criteria which imply that shrinkage estimators of James-Stein type have lower risk than under Euclidean quadratic loss. Further, we show that the corresponding positive-part estimators have lower risk than their unrestricted counterparts, thereby rendering the latter estimators inadmissible. We derive results for the case in which is block-diagonal, the loss function is quadratic and non-spherical, and the shrinkage estimator is constructed by means of a nondecreasing, differentiable function of a quadratic form in . For the problem of shrinking to a vector whose components have a common value constructed from the data, we derive improved shrinkage estimators and again determine conditions under which the positive-part analogs have lower risk than their unrestricted counterparts.     

Robust discrimination under a hierarchy on the scatter matrices

Under normality, Flury and Schmid [Quadratic discriminant functions with constraints on the covariances matrices: some asymptotic results, J. Multivariate Anal. 40 (1992) 244-261] investigated the asymptotic properties of the quadratic discrimination procedure under hierarchical models for the scatter matrices, that is: (i) arbitrary scatter matrices, (ii) common principal components, (iii) proportional scatter matrices and (iv) identical matrices. In this paper, we study the properties of robust quadratic discrimination rules based on robust estimates of the involved parameters. Our analysis is based on the partial influence functions of the functionals related to these parameters and allows to derive the asymptotic variances of the estimated coefficients under models (i)-(iv). From them, we conclude that the asymptotic variances verify the same order relations as those obtained by Flury and Schmid [Quadratic discriminant functions with constraints on the covariances matrices: some asymptotic results, J. Multivariate Anal. 40 (1992) 244-261] for the classical estimators. We also perform a Monte Carlo study for different sample sizes and different hierarchies which shows the advantage of using robust procedures over classical ones, when anomalous data are present. It also confirms that better rates of misclassification can be achieved if a more parsimonious model among all the correct ones is used instead of the standard quadratic discrimination.     

Joint distributions of numbers of occurrences of a discrete pattern and weak convergence of an empirical process for the pattern

For a {0, 1}-pattern of finite length, an empirical process is introduced in order to describe the number of overlapping occurrences of the pattern at each level t[0,1] in a sequence of the corresponding indicators of i.i.d. [0, 1]-valued observations of length n. A method for obtaining the exact finite-dimensional distributions of the empirical process is given. The weak convergence of the process to a Gaussian process in D[0,1] as n tends to infinity is also established. The limiting process depends on the given pattern. The exact covariance function is compared with the asymptotic covariance function in a numerical example.