Multivariate Normal Distribution

The paper addresses asymptotic estimation of normal means under sparsity. The primary focus is estimation of multivariate normal means where we obtain exact asymptotic minimax error under globallocal shrinkage prior. This extends the corresponding univariate work of Ghosh and Chakrabarti (2017). In addition, we obtain similar results for the Dirichlet-Laplace prior as considered in . Also, following van der Pas, Szabó, and van der Vaart (2017), we have been able to derive credible sets for multivariate normal means under global-local priors.

In this paper, two-stage and fixed-sample-size procedures are developed for constructing confidence intervals for the common correlation ρ of an equi-correlated multivariate normal distribution. Two different approaches for estimation are considered, namely fixed-width and bounded-width interval estimation. In the fixed-width confidence interval estimation problem, a two-stage procedure is developed and the exact distribution of the corresponding stopping variable and the exact coverage probability of the interval estimator are derived. Asymptotic optimality properties such as asymptotic first-order efficiency and asymptotic consistency properties of the two-stage procedure are established. Bounded-width confidence intervals are obtained by applying fixed-accuracy estimation methodologies of Mukhopadhyay and Banerjee (2014,2015a,b) and Banerjee and Mukhopadhyay (2016) using different transformations on ρ. Both fixed-sample-size and two-stage sampling methodologies for bounded-width confidence interval estimation of ρ are developed incorporating such transformations. Finally, performances of all the procedures are compared via extensive sets of simulation studies.

We present and study a procedure for testing the null hypothesis of multivariate elliptical symmetry. The procedure is based on the averages of some spherical harmonics over the projections of the scaled residual (1978, N. J. H. Small, Biometrika 65, 657-658) of the d-dimensional data on the unit sphere of R d . We find, under mild hypothesis, the limiting null distribution of the statistic presented, showing that, for an appropriate choice of the spherical harmonics included in the statistic, this distribution does not depend on the parameters that characterize the underlying elliptically symmetric law. We describe a bivariate simulation study that shows that the finite sample quantiles of our statistic converge fairly rapidly, with sample size, to the theoretical limiting quantiles and that our procedure enjoys good power against several alternatives.

By modifying the statistic of , we introduce and study the properties of a notion of multivariate skewness that provides both a magnitude and an overall direction for the skewness present in multivariate data. This notion leads to a test statistic for the null hypothesis of multivariate symmetry. Under mild assumptions, we find the asymptotic distribution of the test statistic and evaluate, by simulations, the convergence of the finite sample size quantiles to their limits, as well as the power of the statistic against some alternatives.

Let X = (X1, X2,..., Xd) ~ be a random vector of positive entries, such that for some )~ = ()h, A2,..., Ad) t, the vector X (~) defined by X~ ~) --(X~ ~ -1)/),~, i = 1,..., d is elliptically symmetric. We describe a procedure based on the multivariate empirical characteristic function for estimating the As's. Asymptotic results regarding consistency of the estimators are given and we evaluate their performance in simulated data. In a one-dimensional setting, comparisons are made with other available transformations to symmetry.

This work describes an unsupervised method to objectively quantify the abnormality of general anatomical shapes. The severity of an anatomical deformity often serves as a determinant in the clinical management of patients. However, experiential bias and distinctive random residuals among specialist individuals bring variability in diagnosis and patient management decisions, irrespective of the objective deformity degree. Therefore, supervised methods are prone to be misled given insufficient labeling of pathological samples that inevitably preserve human bias and inconsistency. Furthermore, subjects demonstrating a specific pathology are naturally rare relative to the normal population. To avoid relying on sufficient pathological samples by fully utilizing the power of normal samples, we propose the shape normality metric (SNM), which requires learning only from normal samples and zero knowledge about the pathology. We represent shapes by landmarks automatically inferred from the data and model the normal group by a multivariate Gaussian distribution. Extensive experiments on different anatomical datasets, including skulls, femurs, scapulae, and humeri, demonstrate that SNM can provide an effective normality measurement, which can significantly detect and indicate pathology. Therefore, SNM offers promising value in a variety of clinical applications.

This work describes an unsupervised method to objectively quantify the abnormality of general anatomical shapes. The severity of an anatomical deformity often serves as a determinant in the clinical management of patients. However, experiential bias and distinctive random residuals among specialist individuals bring variability in diagnosis and patient management decisions, irrespective of the objective deformity degree. Therefore, supervised methods are prone to be misled given insufficient labeling of pathological samples that inevitably preserve human bias and inconsistency. Furthermore, subjects demonstrating a specific pathology are naturally rare relative to the normal population. To avoid relying on sufficient pathological samples by fully utilizing the power of normal samples, we propose the shape normality metric (SNM), which requires learning only from normal samples and zero knowledge about the pathology. We represent shapes by landmarks automatically inferred from the da...

Let us consider (Ω, ℱ, 𝑃) be a probability space of a random experiment. A collection or vector 𝑋 ~ = (𝑋 1 , 𝑋 2 , ⋯ , 𝑋 𝑝 ) 𝑇 defined on (Ω, ℱ, 𝑃) and maps Ω into ℛ 𝑝 i.e., (𝑣𝑖) For all 𝑥 ~= (𝑥 1 , 𝑥 2 , ⋯ , 𝑥 𝑝 ) 𝑇 ∈ ℛ 𝑝 and for all ℎ 𝑖 > 0 ( 𝑖 = 1(1)𝑝), the following inequality holds.

This paper derives the characteristic function of the univariate
t-distributiou in terms of a well-known special function, namely,
the Macdouald function, which has been extensively studied by several authors in recent years. Moments of the distribution have been
found by using the series representation of the characteristic function as well as by an integral representation of the Macdonald function.

In this paper we investigate some of the well known properties of the multivariate t-disribution by directly utilizing the Macdonald function represeatation of its characteristic function. Consequently, we obtain a
limit theorem of the Macdonald function.

In a recent paper Muirhead (1986) derived certaiu useful ideutities iavolviag expectatione taken with respect to the wishart distribution. This uote generalizes the above results by taking expectations with respect to a geaeralized. version of the wishart distributioa, coosidered by Sutradhar and Ali (1ggg), based on a multivariate t-distribution.

We consider n pairs of random variables (Xt1,Xzt),(Xn,Xz),...,(Xr,,Xz) having a bivariate elliptically contoured density of the form K(n)lt\l-'12 s where 01, 02 are location parameters and A : ((1,*)) is a 2 x 2 symmetric positive definite matrix

It is shown (Proposition (3.9)) that the asymptotic information bound which is valid for the estimation of a parameter in the structure (mixture) model remains valid in the functional model (incidental nuisance parameters) if only perturbation symmetric estimators (Definition (3.6)) are admitted. Perturbation symmetry is a property which is closely related to permutation symmetry (Theorem (3.4)). In particular, equicontinuous functions of empirical processes are perturbation symmetric (Theorem (3.3)). Thus, the results of this paper continue a discussion initiated by Bickel and Klaassen, [2], Pfanzagl, [14], and Strasser, [21], on permutation symmetry of estimators and the exclusion of superefficiency in the functional model.

The Fully Bayesian Significance Test (FBST) is a coherent Bayesian significance test for sharp hypotheses. The computation of the evidence measure used on the FBST is performed in two steps: 1) The optimization step consists of finding f * , the maximum (supremum) of the posterior under the null hypothesis. 2) The integration step consists of integrating the posterior density over the Tangential Set, T , where the posterior is higher than f * . This integral is the measure of evidence against the hipothesis H. This paper proposes the FBST as a model selection tool for multivariate normal mixture models, where the problem is to estimate the number of components in the mixture and the parameters for each component. In the FBST formulation of the problem, the base model has m components, and the hypothesis to be tested is the constraint of having m -1 components. The FBST selects the m component model, rejecting H, if the evidence against the hypothesis is above a given threshold τ , and selects the m -1 component model, accepting H, otherwise. We perform two simulated numerical experiments and compare the FBST performance with Mclust, a model-based clustering software. Both experimental tests show that: -the miss rates (proportion of wrong answers) for FBST are significantly lower than Mclust; -the miss rates convergence towards zero for increasing sample sizes is much faster for FBST than Mclust. These results show that FBST is a robust tool for mixture models, and strongly encourage further developments.

We review the definition of the Full Bayesian Significance Test (FBST), and summarize its main statistical and epistemological charac- teristics. We review also the Abstract Belief Calculus (ABC) of Darwiche and Ginsberg, and use it to analyze the FBST's value of evidence. This analysis helps us understand the FBST properties and interpretation. The definition of value of evidence against a sharp hypothesis, in the FBST setup, was moti- vated by applications of Bayesian statistical reasoning to legal matters where the sharp hypotheses were defendants statements, to be judged according to the Onus Probandi ju- ridical principle.

The Full Bayesian Significance Test (FBST) for precise hypotheses is applied to a Multivariate Normal Structure (MNS) model. In the FBST we compute the evidence against the precise hypothesis. This evidence is the probability of the Highest Relative Surprise Set (HRSS) tangent to the sub-manifold (of the parameter space) that defines the null hypothesis. The MNS model we present appears when testing equivalence conditions for genetic expression measurements, using micro-array technology.

Maier et al. (2010) introduced the relational causal model (RCM) for representing and inferring causal relationships in relational data. A lifted representation, called abstract ground graph (AGG), plays a central role in reasoning with and learning of RCM. The correctness of the algorithm proposed by Maier et al. (2013a) for learning RCM from data relies on the soundness and completeness of AGG for relational d-separation to reduce the learning of an RCM to learning of an AGG. We revisit the definition of AGG and show that AGG, as defined in Maier et al. (2013b), does not correctly abstract all ground graphs. We revise the definition of AGG to ensure that it correctly abstracts all ground graphs. We further show that AGG representation is not complete for relational d-separation, that is, there can exist conditional independence relations in an RCM that are not entailed by AGG. A careful examination of the relationship between the lack of completeness of AGG for relational d-separa...

Although the hazard rates sharing a number of similar aspects. The hazard functions are far less frequently used. The problem of hazard functions estimation has received much attention in the statistical literature. This paper studies the hazard rate estimation with the peak over threshold (POT) modeling within completed data. Next, we have to adapt this estimation with the censored data. Ditto, the asymptotic normality has been established, whereas the case of random threshold t under the uncensored and the censored data, In this dissertation, our procedure of our asymptotic result is based on POT context with the threshold t is random. Wherefore we study the asymptotic normality of the proportion of noncensored observation estimator with the threshold t is random. As an application, we present a new extreme value index (EVI) in terms of the hazard functions

In this study we present a closed form solution to the moments and, in particular, correlation of two log-normally distributed random variables, where the underlying log-normal distribution is potentially truncated and censored at both tails. Throughout the analysis we further assume that the parameters of the unconstrained bivariate log-normal distribution are known. The closed form solution also covers the cases where one tail is truncated and the other is censored.

This paper introduces a new family of multivariate distributions based on Gram-Charlier and Edgeworth expansions. This family encompasses many of the univariate seminonparametric densities proposed in the financial econometrics as marginal distributions of the different formulations. Within this family, we focus on the specifications that guarantee positivity so obtaining a well-defined multivariate density. We compare different "positive" multivariate distributions of the family with the multivariate Edgeworth-Sargan, Normal and Student's t in an in-and out-sample framework for financial returns data. Our results show that the proposed specifications provide a quite reasonably good performance being so of interest for applications involving the modelling and forecasting of heavy-tailed distributions.

Numerical computation of multivariate normal probability integrals is often required in quantitative genetic studies. In particular, this is the case for the evaluation of the genetic superiorities after independent culling levels selection on several correlated traits, for certain methods used to analyse discrete traits and for some studies on selection involving a limited number of candidates. Dutt's and Deak's methods can satisfy most of the geneticist's needs. They are presented in this paper and their precision is analysed in detail. It appears that Dutt's method is remarkably precise for dimensions 1 to 5, except when truncation points or correlation coefficients between traits are very high in absolute value. Deak's method, less precise, is better suited for higher dimensions (6 to 20) and more generally for all the situations where Dutt's method is no longer adequate. Key words : Multiple integral, multivaiiate normal distribution, independent culling level selec- tion, multivariate probability integrals. Intérêt en génétique quantitative des méthodes de Dutt et de Deak pour le calcul numérique des intégrales de la loi multinormale Le calcul numérique d'intégrales de lois multinormales est souvent rendu nécessaire dans les études de génétique quantitative : c'est en particulier le cas pour l'évaluation des effets génétiques d'une sélection à niveaux indépendants sur plusieurs caractères corrélés, pour certaines méthodes d'analyse de caractères discontinus ou pour certaines études de sélection portant sur des effectifs limités. Les méthodes de Dutt et de Deak peuvent satisfaire une grande partie des besoins des généticiens. Celles-ci sont présentées dans cet article et leur précision est analysée de façon détaillée. Il apparaît que la méthode de Dutt est remarquablement précise pour les dimensions 1 à 5, sauf lorsque les seuils de troncature ou les corrélations entre variables sont très élevés en valeur absolue. La méthode de Deak, moins précise, convient mieux pour les dimensions supérieures (de 6 à 20) et d'une manière générale pour toutes les situations où la méthode de Dutt est inadéquate. Mots clés : Intégrale multiple, distribution multinormale, sélection à niveaux indépendants.

Bootstrap methods are considered in the application of statistical process control because they can deal with unknown distributions and are easy to calculate using a personal computer. In this study we propose the use of bootstrap-t multivariate control technique on the minimax control chart. The technique takes care of correlated variables as well as the requirement of the distributional assumptions needed for the operation of the minimax control chart. The bootstrap-t technique provides the mean ˆB  of all the bootstrap estimators  is the estimate using the bootstrap sample and B is the number of bootstraps. The computation of the proposed bootstrap-t minimax statistic was performed on the values obtained from the bootstrap estimation. This method was used to determine the position of the four control limits of the minimax control chart. The bootstrap-t approach introduced to minimax multivariate control chart helps to detect shifts in the mean vector of a multivariate process and it overcomes the computational complexity of obtaining the distribution of multivariate data.

Minimax control chart uses the joint probability distribution of the maximum and minimum standardized sample means to obtain the control limits for monitoring purpose. However, the derivation of the joint probability distribution needed to obtain the minimax control limits is complex. In this paper the multivariate normal distribution is integrated numerically using Simpson's one third rule to obtain a non-linear polynomial (NLP) function. This NLP function is then substituted and solved numerically using Newton Raphson method to obtain the control limits for the minimax control chart. The approach helps to overcome the problem of obtaining the joint probability distribution needed for estimating the control limits of both the maximum and the minimum statistic for monitoring multivariate process.

We consider a wide class of stochastic process traffic assignment models that capture the dayto-day evolving interaction between traffic congestion and drivers' information acquisition and choice processes. Such models provide a description of not only transient change and 'steady' behaviour, but also represent additional variability that occurs through probabilistic descriptions. They are therefore highly suited to modelling both the disturbance and subsequent 'drift' of networks that are subject to some systematic change, be that a road closure or capacity reduction, new policy measure or general change in demand patterns. In this paper we derive analytic results to probabilistically capture the nature of the transient effects following such a systematic change. This can be thought of as understanding what happens as a system moves from varying about one equilibrium state to varying about a new equilibrium state. The results capture analytically the changes over time in descriptors of the system, in terms of link flow means, variances and covariances. Formally, the analytic results hold asymptotically as approximations, as we imagine demand increasing in tandem with capacities; however, our interest is in general cases where such tandem increases do not occur, and so we provide conditions under which our approximations are likely to work well. Numerical results of applying the methods are reported on several examples. The quality of the approximations is assessed through comparisons with Monte Carlo simulations from the true underlying process.

We consider a wide class of stochastic process traffic assignment models that capture the dayto-day evolving interaction between traffic congestion and drivers' information acquisition and choice processes. Such models provide a description of not only transient change and 'steady' behaviour, but also represent additional variability that occurs through probabilistic descriptions. They are therefore highly suited to modelling both the disturbance and subsequent 'drift' of networks that are subject to some systematic change, be that a road closure or capacity reduction, new policy measure or general change in demand patterns. In this paper we derive analytic results to probabilistically capture the nature of the transient effects following such a systematic change. This can be thought of as understanding what happens as a system moves from varying about one equilibrium state to varying about a new equilibrium state. The results capture analytically the changes over time in descriptors of the system, in terms of link flow means, variances and covariances. Formally, the analytic results hold asymptotically as approximations, as we imagine demand increasing in tandem with capacities; however, our interest is in general cases where such tandem increases do not occur, and so we provide conditions under which our approximations are likely to work well. Numerical results of applying the methods are reported on several examples. The quality of the approximations is assessed through comparisons with Monte Carlo simulations from the true underlying process.

PurposeThe purpose of this study is to extend the classical noncentral F-distribution under normal settings to noncentral closed skew F-distribution for dealing with independent samples from multivariate skew normal (SN) distributions.Design/methodology/approachBased on generalized Hotelling's T2 statistics, confidence regions are constructed for the difference between location parameters in two independent multivariate SN distributions. Simulation studies show that the confidence regions based on the closed SN model outperform the classical multivariate normal model if the vectors of skewness parameters are not zero. A real data analysis is given for illustrating the effectiveness of our proposed methods.FindingsThis study’s approach is the first one in literature for the inferences in difference of location parameters under multivariate SN settings. Real data analysis shows the preference of this new approach than the classical method.Research limitations/implicationsFor the r...

Despite surface displacements observed by geodesy are linear combinations of slip at faults in an elastic medium, determining the spatial distribution of fault slip remains a ill-posed inverse problem. A widely used approach to circumvent the illness of the inversion is to add regularization constraints in terms of smoothing and/or damping so that the linear system becomes invertible. However, the choice of regularization parameters is often arbitrary, and sometimes leads to significantly different results. Furthermore, the resolution analysis is usually empirical and cannot be made independently of the regularization. The stochastic approach of inverse problems provides a rigorous framework where the a priori information about the searched parameters is combined with the observations in order to derive posterior probabilities of the unkown parameters. Here, I investigate an approach where the prior probability density function (pdf) is a multivariate Gaussian function, with single truncation to impose positivity of slip or double truncation to impose positivity and upper bounds on slip for interseismic modelling. I show that the joint posterior pdf is similar to the linear untruncated Gaussian case and can be expressed as a truncated multivariate normal (TMVN) distribution. The TMVN form can then be used to obtain semi-analytical formulae for the single, 2-D or n-D marginal pdf. The semi-analytical formula involves the product of a Gaussian by an integral term that can be evaluated using recent developments in TMVN probabilities calculations. Posterior mean and covariance can also be efficiently derived. I show that the maximum posterior (MAP) can be obtained using a non-negative least-squares algorithm for the single truncated case or using the bounded-variable least-squares algorithm for the double truncated case. I show that the case of independent uniform priors can be approximated using TMVN. The numerical equivalence to Bayesian inversions using Monte Carlo Markov chain (MCMC) sampling is shown for a synthetic example and a real case for interseismic modelling in Central Peru. The TMVN method overcomes several limitations of the Bayesian approach using MCMC sampling. First, the need of computer power is largely reduced. Second, unlike Bayesian MCMC-based approach, marginal pdf, mean, variance or covariance are obtained independently one from each other. Third, the probability and cumulative density functions can be obtained with any density of points. Finally, determining the MAP is extremely fast.

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.

We present a graph-based technique for estimating sparse covariance matrices and their inverses from high-dimensional data. The method is based on learning a directed acyclic graph (DAG) and estimating parameters of a multivariate Gaussian distribution based on a DAG. For inferring the underlying DAG we use the PC-algorithm [27] and for estimating the DAG-based covariance matrix and its inverse, we use a Cholesky decomposition approach which provides a positive (semi-)definite sparse estimate. We present a consistency result in the high-dimensional framework and we compare our method with the Glasso [12, 8, 2] for simulated and real data.

We propose a new and computationally efficient algorithm for maximizing the observed log-likelihood for a multivariate normal data matrix with missing values. We show that our procedure based on iteratively regressing the missing on the observed variables, generalizes the standard EM algorithm by alternating between different complete data spaces and performing the E-Step incrementally. In this non-standard setup we prove numerical convergence to a stationary point of the observed log-likelihood. For high-dimensional data, where the number of variables may greatly exceed sample size, we add a Lasso penalty in the regression part of our algorithm and perform coordinate descent approximations. This leads to a computationally very attractive technique with sparse regression coefficients for missing data imputation. Simulations and results on four microarray datasets show that the new method often outperforms other imputation techniques as k-nearest neighbors imputation, nuclear norm minimization or a penalized likelihood approach with an 1 -penalty on the inverse covariance matrix.

We propose estimation methods for change points in high-dimensional covariance structures with an emphasis on challenging scenarios with missing values. We advocate three imputation like methods and investigate their implications on common losses used for change point detection. We also discuss how model selection methods have to be adapted to the setting of incomplete data. The methods are compared in a simulation study and applied to a time series from an environmental monitoring system. An implementation of our proposals within the R-package hdcd is available via the Supplementary materials.

We propose estimation methods for change points in high-dimensional covariance structures with an emphasis on challenging scenarios with missing values. We advocate three imputation like methods and investigate their implications on common losses used for change point detection. We also discuss how model selection methods have to be adapted to the setting of incomplete data. The methods are compared in a simulation study and applied to a time series from an environmental monitoring system. An implementation of our proposals within the R-package hdcd is available via the Supplementary materials.

We present a graph-based technique for estimating sparse covariance matrices and their inverses from high-dimensional data. The method is based on learning a directed acyclic graph (DAG) and estimating parameters of a multivariate Gaussian distribution based on a DAG. For inferring the underlying DAG we use the PC-algorithm [27] and for estimating the DAG-based covariance matrix and its inverse, we use a Cholesky decomposition approach which provides a positive (semi-)definite sparse estimate. We present a consistency result in the high-dimensional framework and we compare our method with the Glasso [12, 8, 2] for simulated and real data.

High-breakdown-point estimators of multivariate location and shape matrices, such as the MM-estimator with smooth hard rejection and the Rocke S-estimator, are generally designed to have high efficiency at the Gaussian distribution. However, many phenomena are non-Gaussian, and these estimators can therefore have poor efficiency. This paper proposes a new tunable S-estimator, termed the S-q estimator, for the general class of symmetric elliptical distributions, a class containing many common families such as the multivariate Gaussian, t-, Cauchy, Laplace, hyperbolic, and normal inverse Gaussian distributions. Across this class, the S-q estimator is shown to generally provide higher maximum efficiency than other leading high-breakdown estimators while maintaining the maximum breakdown point. Furthermore, its robustness is demonstrated to be on par with these leading estimators while also being more stable with respect to initial conditions. From a practical viewpoint, these properties make the S-q broadly applicable for practitioners. This is demonstrated with an example application-the minimum-variance optimal allocation of financial portfolio investments.

Many applications in signal processing require the estimation of mean and covariance matrices of multivariate complex-valued data. Often, the data are non-Gaussian and are corrupted by outliers or impulsive noise. To mitigate this, robust estimators are employed. However, existing robust estimation techniques employed in signal processing, such as M-estimators, provide limited robustness in the multivariate case. For this reason, this paper introduces the signal processing community to the highly robust class of multivariate estimators called multivariate S-estimators. The paper extends multivariate S-estimation theory to the complex-valued domain. The theoretical performances of S-estimators are explored and compared with M-estimators through the practical lens of the minimum variance distortionless response (MVDR) beamformer, and the empirical finite-sample performances of the estimators are explored through the practical lens of direction-of-arrival (DOA) estimation using the multiple signal classification (MUSIC) algorithm. INDEX TERMS Complex elliptically symmetric distribution, complex-valued S-estimator, covariance and shape matrix estimation, robust estimation of multivariate location and scatter, Sq-estimator.

A copula is an aggregation function, that can be used as a dependency model. Any multivariate distribution function can be characterized by its marginals and a copula. When introducing imprecision in the modelling of those distribution functions, different solutions are available to aggregate the univariate uncertainty representations into a multivariate one via the copula. We present some of those solutions, and discuss their respective inclusions for special cases: independence, belief functions, necessity functions and p-boxes.

We study the distribution of the maximum likelihood estimate (MLE) in high-dimensional logistic models, where covariates are Gaussian with an arbitrary covariance structure. We prove that in the limit of large problems holding the ratio between the number p of covariates and the sample size n constant, every finite list of MLE coordinates follows a multivariate normal distribution. Concretely, the jth coordinate βj of the MLE is asymptotically normally distributed with mean α β j and standard deviation σ /τ j ; here, β j is the value of the true regression coefficient, and τ j the standard deviation of the jth predictor conditional on all the others. The numerical parameters α > 1 and σ only depend upon the problem dimensionality p/n and the overall signal strength, and can be accurately estimated. Our results imply that the MLE's magnitude is biased upwards and that the MLE's standard deviation is greater than that predicted by classical theory. We present a series of experiments on simulated and real data showing excellent agreement with the theory.

A depth-based rank sum statistic for multivariate data introduced by Liu and Singh [J. Amer. Statist. Assoc. 88 (1993) 252-260] as an extension of the Wilcoxon rank sum statistic for univariate data has been used in multivariate rank tests in quality control and in experimental studies. Those applications, however, are based on a conjectured limiting distribution, provided by Liu and Singh [J. Amer. Statist. Assoc. 88 (1993) 252-260]. The present paper proves the conjecture under general regularity conditions and, therefore, validates various applications of the rank sum statistic in the literature. The paper also shows that the corresponding rank sum tests can be more powerful than Hotelling's T 2 test and some commonly used multivariate rank tests in detecting location-scale changes in multivariate distributions.

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We construct a flexible dynamic linear model for the analysis and prediction of multivariate time series, assuming a two-piece normal initial distribution for the state vector. We derive a novel Kalman filter for this model, obtaining a two components mixture as predictive and filtering distributions. In order to estimate the covariance of the error sequences, we develop a Gibbs-sampling algorithm to perform Bayesian inference. The proposed approach is validated and compared with a Gaussian dynamic linear model in simulations and on a real data set.

We propose and study the class of Box-Cox elliptical distributions. It provides alternative distributions for modeling multivariate positive, marginally skewed and possibly heavy-tailed data. This new class of distributions has as a special case the class of log-elliptical distributions, and reduces to the Box-Cox symmetric class of distributions in the univariate setting. The parameters are interpretable in terms of quantiles and relative dispersions of the marginal distributions and of associations between pairs of variables. The relation between the scale parameters and quantiles makes the Box-Cox elliptical distributions attractive for regression modeling purposes. Applications to data on vitamin intake are presented and discussed.

In this paper we obtain an adjusted version of the likelihood ratio test for errors-in-variables multivariate linear regression models. The error terms are allowed to follow a multivariate distribution in the class of the elliptical distributions, which has the multivariate normal distribution as a special case. We derive a modified likelihood ratio statistic that follows a chi-squared distribution with a high degree of accuracy. Our results generalize those in Melo & Ferrari (Advances in Statistical Analysis, 2010, 94, 75-87) by allowing the parameter of interest to be vector-valued in the multivariate errors-in-variables model. We report a simulation study which shows that the proposed test displays superior finite sample behavior relative to the standard likelihood ratio test.

The problem of reducing the bias of maximum likelihood estimator in a general multivariate elliptical regression model is considered. The model is very flexible and allows the mean vector and the dispersion matrix to have parameters in common. Many frequently used models are special cases of this general formulation, namely: errors-in-variables models, nonlinear mixed-effects models, heteroscedastic nonlinear models, among others. In any of these models, the vector of the errors may have any multivariate elliptical distribution. We obtain the second-order bias of the maximum likelihood estimator, a bias-corrected estimator, and a bias-reduced estimator. Simulation results indicate the effectiveness of the bias correction and bias reduction schemes.