Confidence regions for the common mean vector of several multivariate normal populations

Suppose that there are independent samples available from several multivariate normal populations with the same mean vector m? but possibly different covariance matrices. The problem of developing a confidence region for the common mean vector based on all the samples is considered. An exact confidence region centered at a generalized version of the well-known Graybill-Deal estimator of m? is developed, and a multiple comparison procedure based on this confidence region is outlined. Necessary percentile points for constructing the confidence region are given for the two-sample case. For more than two samples, a convenient method of approximating the percentile points is suggested. Also, a numerical example is presented to illustrate the methods. Further, for the bivariate case, the proposed confidence region and the ones based on individual samples are compared numerically with respect to their expected areas. The numerical results indicate that the new confidence region is preferable to the single-sample versions for practical use.     

Single use conservative confidence regions in multivariate controlled calibration

We consider a multivariate linear model for multivariate controlled calibration, and construct some conservative confidence regions, which are nonempty and invariant under nonsingular transformations. The computation of our confidence region is easier compared to some of the existing procedures. We illustrate the results using two examples. The simulation results show the closeness of the coverage probability of our confidence regions to the assumed confidence level.     

Partitioning k multivariate normal populations according to equivalence with respect to a standard vector

We propose optimal procedures to achieve the goal of partitioning k multivariate normal populations into two disjoint subsets with respect to a given standard vector. Definition of good or bad multivariate normal populations is given according to their Mahalanobis distances to a known standard vector as being small or large. Partitioning k multivariate normal populations is reduced to partitioning k non-central Chi-square or non-central F distributions with respect to the corresponding non-centrality parameters depending on whether the covariance matrices are known or unknown. The minimum required sample size for each population is determined to ensure that the probability of correct decision attains a certain level. An example is given to illustrate our procedures.     

Statistical inference for multivariate partially linear regression models

In this paper we study a class of multivariate partially linear regression models. Various estimators for the parametric component and the nonparametric component are constructed and their asymptotic normality established. In particular, we propose an estimator of the contemporaneous correlation among the multiple responses and develop a test for detecting the existence of such contemporaneous correlation without using any nonparametric estimation. The performance of the proposed estimators and test is evaluated through some simulation studies and an analysis of a real data set is used to illustrate the developed methodology. The Canadian Journal of Statistics 41: 1–22; 2013 © 2013 Statistical Society of Canada     

Characterization of minimax linear estimators in linear regression

Two characterization theorems of the minimax linear estimator (Mile) are proven for the case, where the regression parameter varies only in an arbitrary ellipsoid. Furthermore, the existence, uniqueness and admissibility of Mile are shown. The explicit determination of Mile is carried out for a special case.     

A multivariate version of Gini's rank association coefficient

In this paper, we introduce a multivariate generalization of the population version of Gini's rank association coefficient, giving a response to this open question posed in [4]. We also study some properties of this version, present the corresponding results for the sample statistic, and provide several examples.     

Cross-cumulants measure for independence

We propose a measure for independence of group of random variables, given by a sum of cross-cumulants of a given order n  . A similar measure was known for the case of fourth-order cross-cumulants from the JADE algorithm for ICA (independent component analysis). We derive a formula for its calculation using cumulant tensors. In the case n=4$n=4$ our formula allows efficient calculation of this measure, using cumulant matrices. Much attention is devoted to the case of six-order cross-cumulants, aiming to show that this measure can be calculated using again cumulant matrices.     

Unified treatment of Cramér-Rao bound for the nonregular density functions

This paper extends the idea of Vincze (1978) and unifies the approach for the uniparameter and multiparameter situations for obtaining the Cramér-Rao inequality.     

Burden of bioinformatics in medical research: Statistical perspectives and controversies

The ongoing evolution of genomics and bioinformatics has an overwhelming impact on medical and clinical research, albeit this development is often marked by genuine controversies as well as lack of scientific clarities and acumen. The search for disease genes and the gene–environment interaction has drawn considerable interdisciplinary scientific attention: environmental health, clinical and medical sciences, biological as well as computational and statistical sciences are most noteworthy. Statistical reasoning (quantitative modeling and analysis perspectives) has a focal stand in this respect while data mining resolutions are far from being statistically fully understood or interpretable. The use of human subjects, though unavoidable, under various extraneous restraints, medical ethics perspectives, and human rights undercurrents, has raised concern all over the world, especially in the developing countries. In the genomics context, clinical trials may be designed on chips and yet there are greater challenges due to the curse of dimensionality perspectives. Some of these challenging statistical issues in medical and clinical research (with emphasis on clinical trials) are appraised in the light of existing statistical tools, which are available for less complex clinical research problems.     

Pseudo confidence regions for the solution of a multivariate linear functional relationship

The problem of finding confidence regions (CR) for a q-variate vector γ given as the solution of a linear functional relationship (LFR) Λγ = μ is investigated. Here an m-variate vector μ and an m × q matrix Λ = (Λ₁, Λ₂,…, Λ_q) are unknown population means of an m(q+1)-variate normal distribution $\text{N}{}_{\mathrm{m(q+1)}}\text{(ζΩ?Σ)}$, where ζ′ = (μ′, Λ₁′, Λ₂′,…, Λ_q′Σ is an unknown, symmetric and positive definite m × m matrix and Ω is a known, symmetric and positive definite (q+1) × (q+1) matrix and ? denotes the Kronecker product. This problem is a generalization of the univariate special case for the ratio of normal means.A CR for γ with level of confidence 1 ? α, is given by a quadratic inequality, which yields the so-called ‘pseudo’ confidence regions (PCR) valid conditionally in subsets of the parameter space. Our discussion is focused on the ‘bounded pseudo’ confidence region (BPCR) given by the interior of a hyperellipsoid. The two conditions necessary for a BPCR to exist are shown to be the consistency conditions concerning the multivariate LFR. The probability that these conditions hold approaches one under ‘reasonable circumstances’ in many practical situations. Hence, we may have a BPCR with confidence approximately 1 ? α. Some simulation results are presented.     

Minimum phi-divergence estimators for loglinear models with linear constraints and multinomial sampling

In this paper the family ofφ-divergence estimators for loglinear models with linear constraints and multinomial sampling is studied. This family is an extension of the maximum likelihood estimator studied by Haber and Brown (1986). A simulation study is presented and some alternative estimators to the maximum likelihood are obtained. This work was parcially supported by Grant DGES PB2003-892     

Moments of Special Normally Distributed Matrices

In solving systems of simultaneous random linear algebraic equations some approximating methods lead to the problem of determinating moments of special random matrices and vectors. In this article corresponding formulas are provided for moments of some normally distributed matrices. The deduced relations can be considered as a generalization of the known formulas for the central moments of normally distributed random variables.     

Asymptotics for tests on mean profiles,additional information and dimensionality under non-normality

We consider the comparison of mean vectors for k groups when k is large and sample size per group is fixed. The asymptotic null and non-null distributions of the normal theory likelihood ratio, Lawley–Hotelling and Bartlett–Nanda–Pillai statistics are derived under general conditions. We extend the results to tests on the profiles of the mean vectors, tests for additional information (provided by a sub-vector of the responses over and beyond the remaining sub-vector of responses in separating the groups) and tests on the dimension of the hyperplane formed by the mean vectors. Our techniques are based on perturbation expansions and limit theorems applied to independent but non-identically distributed sequences of quadratic forms in random matrices. In all these four MANOVA problems, the asymptotic null and non-null distributions are normal. Both the null and non-null distributions are asymptotically invariant to non-normality when the group sample sizes are equal. In the unbalanced case, a slight modification of the test statistics will lead to asymptotically robust tests. Based on the robustness results, some approaches for finite approximation are introduced. The numerical results provide strong support for the asymptotic results and finiteness approximations.     

Consistency of the least-squares-estimators for linear stochastic difference equations

Assumptions are given for the strong consistency in the stable case and weak consistency in the instable case of the Least-Square-Estimator of the unknown system-parameters of a inhomogeneous linear stochastic difference equation system with constant coefficients.     

A generalization of an identity of hoeffding and some applications

Summary We generalize a well-known identity due to Hoeffding and use this generalization to prove a result of Cambanis, Simons and Stout under somewhat different hypotheses and to extend some results of Lehmann concerning bivariate distributions with quadrant dependence.     

Statistical inference for a semiparametric measurement error regression model with heteroscedastic errors

Efficient inference for regression models requires that the heteroscedasticity be taken into account. We consider statistical inference under heteroscedasticity in a semiparametric measurement error regression model, in which some covariates are measured with errors. This paper has multiple components. First, we propose a new method for testing the heteroscedasticity. The advantages of the proposed method over the existing ones are that it does not need any nonparametric estimation and does not involve any mismeasured variables. Second, we propose a new two-step estimator for the error variances if there is heteroscedasticity. Finally, we propose a weighted estimating equation-based estimator (WEEBE) for the regression coefficients and establish its asymptotic properties. Compared with existing estimators, the proposed WEEBE is asymptotically more efficient, avoids undersmoothing the regressor functions and requires less restrictions on the observed regressors. Simulation studies show that the proposed test procedure and estimators have nice finite sample performance. A real data set is used to illustrate the utility of our proposed methods.     

Improving on the sample covariance matrix for a complex elliptically contoured distribution

In this paper the problem of estimating the scale matrix in a complex elliptically contoured distribution (complex ECD) is addressed. An extended Haff–Stein identity for this model is derived. It is shown that the minimax estimators of the covariance matrix obtained under the complex normal model remain robust under the complex ECD model when the Stein loss function is employed.