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Multivariate normal/independent (MNI) distributions contain many renowned heavy-tailed distributions such as the multivariate t , multivariate slash, multivariate contaminated normal, multivariate variance-gamma, and multivariate double exponential distributions. A frequent problem encountered in statistical analysis is the occurrence of truncated observations and non-normality such that theoretical moments are required for the estimation of the truncated multivariate normal/independent (TMNI) distributions. This paper is dedicated to deriving explicit expressions for the moments of the TMNI distributions with supports confined within a hyper-rectangle. A Monte Carlo experiment is undertaken to validate to the correctness of the proposed formulae for five selected members of the TMNI distributions. R scripts and data to reproduce the results are available in the GitHub repository. [ABSTRACT FROM AUTHOR]

The space of discrete copulas admits a representation as a convex polytope, and this has been exploited in entropy-copula methods used in hydrology and climatology. In this paper, we focus on the class of component-wise convex copulas, i.e., ultramodular copulas, which describe the joint behavior of stochastically decreasing random vectors. We show that the family of ultramodular discrete copulas and its generalization to component-wise convex discrete quasi-copulas also admit representations as polytopes. In doing so, we draw connections to the Birkhoff polytope, the alternating sign matrix polytope, and their generalizations, thereby unifying and extending results on these polytopes from both the statistics and the discrete geometry literature. [ABSTRACT FROM AUTHOR]

Abstract Hypothesis testing for the proportionality of covariance matrices is a classical statistical problem and has been widely studied in the literature. However, there have been few treatments of this test in high-dimensional settings, especially for the case where the number of variables is larger than the sample size, despite high-dimensional statistical inference having recently received considerable attention. This paper studies hypothesis testing for the proportionality of two covariance matrices in the high-dimensional setting: m , n ≍ p δ for some δ ∈ (1 ∕ 2 , 1) , where m and n denote the sample sizes and p denotes the number of variables. A test statistic is proposed and its asymptotic distribution is derived under multivariate normality. The non-asymptotic performance of the proposed test procedure is numerically examined. [ABSTRACT FROM AUTHOR]

The partially linear model (PLM) is a useful semiparametric extension of the linear model that has been well studied in the statistical literature. This paper proposes a variable selection procedure for the PLM with ultrahigh dimensional predictors. The proposed method is different from the existing penalized least squares procedure in that it relies on partial correlation between the partial residuals of the response and the predictors. We systematically study the theoretical properties of the proposed procedure and prove its model consistency property. We further establish the root- n convergence of the estimator of the regression coefficients and the asymptotic normality of the estimate of the baseline function. We conduct Monte Carlo simulations to examine the finite-sample performance of the proposed procedure and illustrate the proposed method with a real data example. [ABSTRACT FROM AUTHOR]

This paper aims to present the various contributions to the Special Issue of the Journal of Multivariate Analysis on Functional Data Analysis and some related topics including High-Dimensional Statistics and Multivariate Analysis of complex data. The presentation is made by emphasizing on how the contributions are behaving among recent trends in the fields, in such a way that this paper can also be viewed as a selected bibliographical discussion about recent advances in these topics. [ABSTRACT FROM AUTHOR]

Abstract: Recently Lao and Mayer (2008) considered -max-statistics, where the maximum of kernels over the set of indices is studied instead of the usual sums. Such statistics emerge frequently in stochastic geometry. The examples include the largest distance between random points in a ball, the maximal diameter of a random polygon, the largest scalar product within a sample of points, etc. Their limit distributions are related to the distributions of extreme values. Among the results obtained by Lao and Mayer, the limit theorems for the maximal perimeter and the maximal area of random triangles inscribed in a circumference are of great interest. In the present paper, we generalize these theorems to the case of convex -polygons, , with random vertices on the circumference. In addition, a similar problem for the minimal perimeter and the minimal area of circumscribed -polygons is solved in this paper. This problem has not been studied in the literature so far. [Copyright &y& Elsevier]

It is well known that, in the presence of outliers, the single linkage algorithm generally fails to identify clusters. In this paper, we construct a new version of this algorithm, less sensitive to outliers, and study both its theoretical properties and its practical behavior. In particular, we provide an oracle-type inequality which guarantees that our procedure recovers clusters with high probability under mild assumptions on the distribution of the outliers. Using this inequality, we prove the consistency of our method and exhibit rates of convergence in various situations. The performance of this approach is also assessed through simulation studies. A thorough comparison with several classical clustering algorithms on simulated data is presented. [ABSTRACT FROM AUTHOR]

Dimension reduction provides a useful tool for statistical data analysis with high-dimensional data. In this paper, we develop a parsimonious multivariate spatial regression model with a non-separable covariance function. The efficacy of this new solution is illustrated through simulation studies and a real data analysis. We show that for cases where the marginal spatial correlations are different from each other, the proposed non-separable model provides better estimation and inference than the related separable model, and provides tighter inference than a non-separable spatial model without dimension reduction when there is immaterial variation in the data. [ABSTRACT FROM AUTHOR]

We consider the edge statistics of large dimensional deformed rectangular matrices of the form Y t = Y + t X , where Y is a p × n deterministic signal matrix whose rank is comparable to n , X is a p × n random noise matrix with i.i.d. entries of mean zero and variance n − 1 , and t > 0 gives the noise level. This model is referred to as the interference-plus-noise matrix in the study of massive multiple-input multiple-output (MIMO) system, which belongs to the category of the so-called signal-plus-noise model. For the case t = 1 , the spectral statistics of this model have been studied to a certain extent in the literature (Dozier and Silverstein, 2007[17,18]; Vallet et al., 2012). In this paper, we study the singular value and singular vector statistics of Y t around the right-most edge of the spectrum in the harder regime n − 1 / 3 ≪ t ≪ 1. This regime is harder than the t = 1 case, because on the one hand, the edge behavior of the empirical spectral distribution (ESD) of Y Y ⊤ has a strong effect on the edge statistics of Y t Y t ⊤ for a "small" t ≪ 1 , while on the other hand, the edge eigenvalue behavior of Y t Y t ⊤ is not merely a perturbation of that of Y Y ⊤ for a "large" t ≫ n − 1 / 3 . Under certain regularity assumptions on Y , we prove the Tracy–Widom law for the edge eigenvalues, the eigenvalue rigidity, and eigenvector delocalization for the matrices Y t Y t ⊤ and Y t ⊤ Y t . These results can be used to estimate and infer the massive MIMO system. To prove the main results, we analyze the edge behavior of the asymptotic ESD of Y t Y t ⊤ and establish optimal local laws on its resolvent. These results are of independent interest, and can be used as important inputs for many other problems regarding the spectral statistics of Y t. [ABSTRACT FROM AUTHOR]

Abstract: This paper analyzes a data mining/bump hunting technique known as PRIM . PRIM finds regions in high-dimensional input space with large values of a real output variable. This paper provides the first thorough study of statistical properties of PRIM. Amongst others, we characterize the output regions PRIM produces, and derive rates of convergence for these regions. Since the dimension of the input variables is allowed to grow with the sample size, the presented results provide some insight about the qualitative behavior of PRIM in very high dimensions. Our investigations also reveal some shortcomings of PRIM, resulting in some proposals for modifications. [Copyright &y& Elsevier]

Abstract: There is a recent interest in developing new statistical methods to predict time series by taking into account a continuous set of past values as predictors. In this functional time series prediction approach, we propose a functional version of the partial linear model that allows both to consider additional covariates and to use a continuous path in the past to predict future values of the process. The aim of this paper is to present this model, to construct some estimates and to look at their properties both from a theoretical point of view by means of asymptotic results and from a practical perspective by treating some real data sets. Although the literature on the use of parametric or nonparametric functional modeling is growing, as far as we know, this is the first paper on semiparametric functional modeling for the prediction of time series. [Copyright &y& Elsevier]

High-throughput expression profiling techniques bring novel tools and also statistical challenges to genetic research. In addition to detecting differentially expressed genes, testing the significance of gene sets or pathway analysis has been recognized as an equally important problem. Owing to the “large p small n ” paradigm, the traditional Hotelling’s T 2 test suffers from the singularity problem and therefore is not valid in this setting. In this paper, we propose a shrinkage-based diagonal Hotelling’s test for both one-sample and two-sample cases. We also suggest several different ways to derive the approximate null distribution under different scenarios of p and n for our proposed shrinkage-based test. Simulation studies show that the proposed method performs comparably to existing competitors when n is moderate or large, but it is better when n is small. In addition, we analyze four gene expression data sets and they demonstrate the advantage of our proposed shrinkage-based diagonal Hotelling’s test. [ABSTRACT FROM AUTHOR]

The Matérn family of covariance functions has played a central role in spatial statistics for decades, being a flexible parametric class with one parameter determining the smoothness of the paths of the underlying spatial field. This paper proposes a family of spatial covariance functions, which stems from a reparameterization of the generalized Wendland family. As for the Matérn case, the proposed family allows for a continuous parameterization of the smoothness of the underlying Gaussian random field, being additionally compactly supported. More importantly, we show that the proposed covariance family generalizes the Matérn model which is attained as a special limit case. This implies that the (reparametrized) Generalized Wendland model is more flexible than the Matérn model with an extra-parameter that allows for switching from compactly to globally supported covariance functions. Our numerical experiments elucidate the speed of convergence of the proposed model to the Matérn model. We also inspect the asymptotic distribution of the maximum likelihood method when estimating the parameters of the proposed covariance models under both increasing and fixed domain asymptotics. The effectiveness of our proposal is illustrated by analyzing a georeferenced dataset of mean temperatures over a region of French, and performing a re-analysis of a large spatial point referenced dataset of yearly total precipitation anomalies. [ABSTRACT FROM AUTHOR]

Abstract: Heterogeneous data are common in social, educational, medical and behavioral sciences. Recently, finite mixture structural equation models (SEMs) and two-level SEMs have been respectively proposed to analyze different kinds of heterogeneous data. Due to the complexity of these two kinds of SEMs, model comparison is difficult. For instance, the computational burden in evaluating the Bayes factor is heavy, and the Deviance Information Criterion may not be appropriate for mixture SEMs. In this paper, a Bayesian criterion-based method called the measure, which involves a component related to the variability of the prediction and a component related to the discrepancy between the data and the prediction, is proposed. Moreover, the calibration distribution is introduced for formal comparison of competing models. Two simulation studies, and two applications based on real data sets are presented to illustrate the satisfactory performance of the measure in model comparison. [Copyright &y& Elsevier]

Abstract: In this paper we study -statistics with side information incorporated using the method of empirical likelihood. Some basic properties of the proposed statistics are investigated. We find that by implementing the side information properly, the proposed -statistics can have smaller asymptotic variance than the existing -statistics in the literature. The proposed -statistics can achieve asymptotic efficiency in a formal sense and their weak limits admit a convolution result. We also find that the corresponding -likelihood ratio procedure, as well as the -empirical likelihood based confidence interval construction, do not benefit from incorporating side information, a result that is consistent with the result under the standard empirical likelihood ratio procedure. The impact of incorrect side information implementation in the proposed -statistics is also explored. Simulation studies are conducted to assess the finite sample performance of the proposed method. The numerical results show that with side information implemented, the deduction of asymptotic variance can be substantial in some cases, and the coverage probability of the confidence interval using the -empirical likelihood ratio based method outperforms that of the normal approximation based method, in particular in the cases when the underlying distribution is skewed. [Copyright &y& Elsevier]

Abstract: Validity of Wagle’s multivariate extension of the Durbin randomization device is directly proved and some vague points of the original paper are clarified. The device is then used in a non-standard way to obtain asymptotic distributions of some functions of the multivariate Gaussian sample configuration. Applications comprise, e.g., null distributions of some statistics naturally emerging in the context of invariant testing multivariate normality. [Copyright &y& Elsevier]

Abstract: Minimum distance techniques have become increasingly important tools for solving statistical estimation and inference problems. In particular, the successful application of the Hellinger distance approach to fully parametric models is well known. The corresponding optimal estimators, known as minimum Hellinger distance estimators, achieve efficiency at the model density and simultaneously possess excellent robustness properties. For statistical models that are semiparametric, in that they have a potentially infinite dimensional unknown nuisance parameter, minimum distance methods have not been fully studied. In this paper, we extend the Hellinger distance approach to general semiparametric models and study minimum Hellinger distance estimators for semiparametric models. Asymptotic properties such as consistency, asymptotic normality, efficiency and adaptivity of the proposed estimators are investigated. Small sample and robustness properties of the proposed estimators are also examined using a Monte Carlo study. Two real data examples are analyzed as well. [Copyright &y& Elsevier]

Abstract: The size-and-shape and shape distributions based on non-central and non-isotropic elliptical distributions are derived in this paper by using the singular value decomposition (SVD). The general densities require the computation of new integrals involving zonal polynomials. The invariance of the central shape distribution is also proved. Finally, some particular densities are applied in a classical data of Biology, and the inference based on exact distributions is performed after choosing the best model by using a modified criterion. [Copyright &y& Elsevier]

Abstract: An important model in handling the multivariate data is the partially linear single-index regression model with a very flexible distribution—beta distribution, which is commonly used to model data restricted to some open intervals on the line. In this paper, the score test is extended to the partially linear single-index beta regression model. The penalized likelihood estimation based on P-spline is proposed. Based on the estimation, the score test statistics about varying dispersion parameter is given. Its asymptotical property is investigated. Both simulated examples are used to illustrate our proposed methods. [Copyright &y& Elsevier]

Abstract: In this article we shall consider a class of strongly -periodically correlated processes with values in a separable complex Hilbert space . The periodograms of these processes and their statistical properties were previously studied by the authors. In this paper we derive the asymptotic distribution of the periodogram, that appears to be a certain Wishart distribution on . [Copyright &y& Elsevier]

Abstract: Data are often affected by uncertainty. Uncertainty is usually referred to as randomness. Nonetheless, other sources of uncertainty may occur. In particular, the empirical information may also be affected by imprecision. Also in these cases it can be fruitful to analyze the underlying structure of the data. In this paper we address the problem of summarizing a sample of three-way imprecise data. In order to manage the different sources of uncertainty a twofold strategy is adopted. On the one hand, imprecise data are transformed into fuzzy sets by means of the so-called fuzzification process. The so-obtained fuzzy data are then analyzed by suitable generalizations of the Tucker3 and CANDECOMP/PARAFAC models, which are the two most popular three-way extensions of Principal Component Analysis. On the other hand, the statistical validity of the obtained underlying structure is evaluated by (nonparametric) bootstrapping. A simulation experiment is performed for assessing whether the use of fuzzy data is helpful in order to summarize three-way uncertain data. Finally, to show how our models work in practice, an application to real data is discussed. [Copyright &y& Elsevier]

Abstract: The theory of quasi-arithmetic means represents a powerful tool in the study of covariance functions across space–time. In the present study we use quasi-arithmetic functionals to make inferences about the permissibility of averages of functions that are not, in general, permissible covariance functions. This is the case, e.g., of the geometric and harmonic averages, for which we obtain permissibility criteria. Also, some important inequalities involving covariance functions and preference relations as well as algebraic properties can be derived by means of the proposed approach. In particular, quasi-arithmetic covariances allow for ordering and preference relations, for a Jensen-type inequality and for a minimal and maximal element of their class. The general results shown in this paper are then applied to the study of spatial and spatio-temporal random fields. In particular, we discuss the representation and smoothness properties of a weakly stationary random field with a quasi-arithmetic covariance function. Also, we show that the generator of the quasi-arithmetic means can be used as a link function in order to build a space–time nonseparable structure starting from the spatial and temporal margins, a procedure that is technically sound for those working with copulas. Several examples of new families of stationary covariances obtainable with this procedure are shown. Finally, we use quasi-arithmetic functionals to generalise existing results concerning the construction of nonstationary spatial covariances, and discuss the applicability and limits of this generalisation. [Copyright &y& Elsevier]

Abstract: A simultaneous confidence band provides useful information on the plausible range of the unknown regression model, and different confidence bands can often be constructed for the same regression model. For a simple regression line, Liu and Hayter [W. Liu, A.J. Hayter, Minimum area confidence set optimality for confidence bands in simple linear regression, J. Amer. Statist. Assoc. 102 (477) (2007) pp. 181–190] proposed the use of the area of the confidence set corresponding to a confidence band as an optimality criterion in comparison of confidence bands; the smaller the area of the confidence set, the better the corresponding confidence band. This minimum area confidence set (MACS) criterion can be generalized to a minimum volume confidence set (MVCS) criterion in the study of confidence bands for a multiple linear regression model. In this paper hyperbolic and constant width confidence bands for a multiple linear regression model over a particular ellipsoidal region of the predictor variables are compared under the MVCS criterion. It is observed that whether one band is better than the other depends on the magnitude of one particular angle that determines the size of the predictor variable region. When the angle and hence the size of the predictor variable region is small, the constant width band is better than the hyperbolic band but only marginally. When the angle and hence the size of the predictor variable region is large the hyperbolic band can be substantially better than the constant width band. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a semiparametric regression model where the unknown regression function is the sum of parametric and nonparametric parts. The parametric part is a finite-dimensional multiple regression function whereas the nonparametric part is represented by an infinite series of orthogonal basis. In this model, we investigate the large sample property of the Bayes factor for testing the parametric null model against the semiparametric alternative model. Under some conditions on the prior and design matrix, we identify the analytic form of the Bayes factor and show that the Bayes factor is consistent, i.e. converges to infinity in probability under the parametric null model, while converges to zero under the semiparametric alternative, as the sample size increases. [Copyright &y& Elsevier]

Abstract: Motivated by a practical problem, [Z.W. Cai, P.A. Naik, C.L. Tsai, De-noised least squares estimators: An application to estimating advertising effectiveness, Statist. Sinica 10 (2000) 1231–1243] proposed a new regression model with noised variables due to measurement errors. In this model, the means of some covariates are nonparametric functions of an auxiliary variable. They also proposed a de-noised estimator for the parameters of interest, and showed that it is root- consistent and asymptotically normal when undersmoothing is applied. The undersmoothing, however, causes difficulty in selecting the bandwidth. In this paper, we propose an alternative corrected de-noised estimator, which is asymptotically normal without the need for undersmoothing. The asymptotic normality holds over a fairly wide range of bandwidth. A consistent estimator of the asymptotic covariance matrix under a general stationary error process is also proposed. In addition, we discuss the fitting of the error structure, which is important for modeling diagnostics and statistical inference, and extend the existing error structure fitting method to this new regression model. A simulation study is made to evaluate the proposed estimators, and an application to a set of advertising data is also illustrated. [Copyright &y& Elsevier]

Abstract: Consider the model , where is unknown but smooth. It is often assumed that and are independent. However, in practice this assumption is violated in many cases. In this paper we propose modeling the dependence between and by means of a copula model, i.e. , where is a copula function depending on an unknown parameter , and and are the marginals of and . Since many parametric copula families contain the independent copula as a special case, the so-obtained regression model is more flexible than the ‘classical’ regression model. We estimate the parameter via a pseudo-likelihood method and prove the asymptotic normality of the estimator, based on delicate empirical process theory. We also study the estimation of the conditional distribution of given . The procedure is illustrated by means of a simulation study, and the method is applied to data on food expenditures in households. [Copyright &y& Elsevier]

Abstract: Multivariate mode hunting is of increasing practical importance. Only a few such methods exist, however, and there usually is a trade-off between practical feasibility and theoretical justification. In this paper we attempt to do both. We propose a method for locating isolated modes (or better, modal regions) in a multivariate data set without pre-specifying their total number. Information on significance of the findings is provided by means of formal testing for the presence of antimodes. Critical values of the tests are derived from large sample considerations. The method is designed to be computationally feasible in moderate dimensions, and it is complemented by diagnostic plots. Since the null hypothesis under consideration is highly composite the proposed tests involve calibration in order to ensure a correct (asymptotic) level. Our methods are illustrated by application to real data sets. [Copyright &y& Elsevier]

Abstract: Let be a convex mapping and a Hilbert space. In this paper we prove the following refinement of Jensen’s inequality: for every such that and . Expectations of Hilbert-space-valued random elements are defined by means of the Pettis integrals. Our result generalizes a result of [S. Karlin, A. Novikoff, Generalized convex inequalities, Pacific J. Math. 13 (1963) 1251–1279], who derived it for . The inverse implication is also true if is an absolutely continuous probability measure. A convexity criterion based on the Jensen-type inequalities follows and we study its asymptotic accuracy when the empirical distribution function based on an -dimensional sample approximates the unknown distribution function. Some statistical applications are addressed, such as nonparametric estimation and testing for convex regression functions or other functionals. [Copyright &y& Elsevier]

Abstract: Let be independent exponential random variables with respective hazard rates , and let be independent exponential random variables with common hazard rate . This paper proves that , the second order statistic of , is larger than , the second order statistic of , in terms of the likelihood ratio order if and only if with . Also, it is shown that is smaller than in terms of the likelihood ratio order if and only if These results form nice extensions of those on the hazard rate order in Paˇltaˇnea [E. Paˇltaˇnea, On the comparison in hazard rate ordering of fail-safe systems, Journal of Statistical Planning and Inference 138 (2008) 1993–1997]. [Copyright &y& Elsevier]

Abstract: The fitting of finite mixture models is an ill-defined estimation problem, as completely different parameterizations can induce similar mixture distributions. This leads to multiple modes in the likelihood, which is a problem for frequentist maximum likelihood estimation, and complicates statistical inference of Markov chain Monte Carlo draws in Bayesian estimation. For the analysis of the posterior density of these draws, a suitable separation into different modes is desirable. In addition, a unique labelling of the component specific estimates is necessary to solve the label switching problem. This paper presents and compares two approaches to achieve these goals: relabelling under multimodality and constrained clustering. The algorithmic details are discussed, and their application is demonstrated on artificial and real-world data. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to present a framework for asymptotic analysis of likelihood ratio and minimum discrepancy test statistics. First order asymptotics are presented in a general framework under minimal regularity conditions and for not necessarily nested models. In particular, these asymptotics give sufficient and in a sense necessary conditions for asymptotic normality of test statistics under alternative hypotheses. Second order asymptotics, and their implications for bias corrections, are also discussed in a somewhat informal manner. As an example, asymptotics of test statistics in the analysis of covariance structures are discussed in detail. [Copyright &y& Elsevier]

Abstract: In this paper, we define two restricted estimators for the regression parameters in a multiple linear regression model with measurement errors when prior information for the parameters is available. We then construct two sets of improved estimators which include the preliminary test estimator, the Stein-type estimator and the positive rule Stein type estimator for both slope and intercept, and examine their statistical properties such as the asymptotic distributional quadratic biases and the asymptotic distributional quadratic risks. We remove the distribution assumption on the error term, which was generally imposed in the literature, but provide a more general investigation of comparison of the quadratic risks for these estimators. Simulation studies illustrate the finite-sample performance of the proposed estimators, which are then used to analyze a dataset from the Nurses Health Study. [Copyright &y& Elsevier]

Abstract: Through this paper it is shown that if the Tukey depths of two probabilities, and , coincide and one of those distributions is discrete, then . The same is proved if the random Tukey depths coincide. [Copyright &y& Elsevier]

Abstract: In this paper, an information-based criterion is proposed for carrying out change point analysis and variable selection simultaneously in linear models with a possible change point. Under some weak conditions, this criterion is shown to be strongly consistent in the sense that with probability one, it chooses the smallest true model for large . Its byproducts include strongly consistent estimates of the regression coefficients regardless if there is a change point. In case that there is a change point, its byproducts also include a strongly consistent estimate of the change point parameter. In addition, an algorithm is given which has significantly reduced the computation time needed by the proposed criterion for the same precision. Results from a simulation study are also presented. [Copyright &y& Elsevier]

Abstract: This paper studies case deletion diagnostics for multilevel models. Using subset deletion, diagnostic measures for identifying influential units at any level are developed for both fixed and random parameters. Two approximate update formulae are derived. The first formula uses one-step approximation, while the second formula also includes the impact of estimating the random parameter. Two examples are used to illustrate the methodology developed. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: Autoregressive time series models of order p have parameters, the mean, the variance of the white noise and the p autoregressive parameters. Change in any of these over time is a sign of disturbance that is important to detect. The methods of this paper can test for change in any one of these parameters separately, or in any collection of them. They are available in forms that make one-sided tests possible, furthermore, they can be used to test for a temporary change. The test statistics are based on the efficient score vector. The large sample properties of the change-point estimator are also explored. [Copyright &y& Elsevier]

Abstract: We consider a class of statistics based on -divergence for the test of independence in contingency tables. The class of statistics includes the statistics based on the power divergence as a special case. Statistic is the log likelihood ratio statistic and is Pearson''s statistic. Statistic corresponds to the statistic recommended by Cressie and Read [Multinomial goodness-of-fit tests, J. Roy. Statist. Soc. B 46 (1984) 440–464] for the goodness-of-fit test. All members of statistics have the same chi-square limiting distribution under the hypothesis of independence. In this paper, we show the derivation of an expression of approximation for the distribution of under the hypothesis of independence. The expression consists of continuous and discontinuous terms. Using the continuous term of the expression, we propose a new approximation of the distribution of . Furthermore, on the basis of the approximation, we obtain transformations that improve the speed of convergence to the chi-square limiting distribution of . As a competitor of the transformed statistic, we derive a moment-corrected-type statistic. By numerical comparison in the case of , we show that the transformed statistic performs very well. [Copyright &y& Elsevier]

Abstract: In this paper we provide rather weak conditions on a distribution which would guarantee that the t-statistic of a random vector of order n follows the t-distribution with degrees of freedom. The results sharpen the earlier conclusions of Mauldon [Characterizing properties of statistical distributions, Quart. J. Math. 2(7) (1956) 155–160] and the more recent advances due to Bondesson [When is the t-statistic t-distributed, Sankhyā, Ser. A 45 (1983) 338–345]. The basic tool involved in the derivations is the vertical density representation originally suggested by Troutt [A theorem on the density of the density ordinate and an alternative interpretation of the Box–Muller method, Statistics 22(3) (1991) 463–466; Vertical density representation and a further remark on the Box–Muller method, Statistics 24 (1993) 81–83]. Several illustrative examples are presented. [Copyright &y& Elsevier]

Abstract: Li et al. [Distributions with Fixed Marginals and Related Topics, vol. 28, Institute of Mathematics and Statistics, Hayward, CA, 1996, pp. 198–212] provide bounds on the distribution and on the tail for functions of dependent random vectors having fixed multivariate marginals. In this paper, we correct a result stated in the above article and we give improved bounds in the case of the sum of identically distributed random vectors. Moreover, we provide the dependence structures meeting the bounds when the fixed marginals are uniformly distributed on the k-dimensional hypercube. Finally, a definition of a multivariate risk measure is given along with actuarial/financial applications. [Copyright &y& Elsevier]

Abstract: In logistic case–control studies, Prentice and Pyke (Biometrika 66 (1979) 403–411) showed that valid point estimators of the odds-ratio parameters and their standard errors may be obtained by fitting the prospective logistic regression model to case–control data. Wang and Carroll (Biometrika 80 (1993) 237–241; J. Statist. Plann. Inference 43 (1995) 331–340) generalized Prentice and Pyke''s (Biometrika 66 (1979) 403–411) results to robust logistic case–control studies. In this paper, we extend the results of Prentice and Pyke (Biometrika 66 (1979) 403–411) and Wang and Carroll (Biometrika 80 (1993) 237–241; J. Statist. Plann. Inference 43 (1995) 331–340) to a class of statistics and a class of unbiased estimating equations. We present some results on simulation and on the analysis of two real datasets. [Copyright &y& Elsevier]

Semiparametric models to describe the functional relationship between k groups of observations are broadly applied in statistical analysis, ranging from nonparametric ANOVA to proportional hazard (ph) rate models in survival analysis. In this paper we deal with the empirical assessment of the validity of such a model, which will be denoted as a “structural relationship model”. To this end Hadamard differentiability of a suitable goodness-of-fit measure in the k-sample case is proved. This yields asymptotic limit laws which are applied to construct tests for various semiparametric models, including the Cox ph model. Two types of asymptotics are obtained, first when the hypothesis of the semiparametric model under investigation holds true, and second for the case when a fixed alternative is present. The latter result can be used to validate the presence of a semiparametric model instead of simply checking the null hypothesis “the model holds true”. Finally, various bootstrap approximations are numerically investigated and a data example is analyzed. [Copyright &y& Elsevier]

Abstract: This paper is primarily concerned with the open problem of minimizing the lower tail of the multinomial distribution. During the study of that specific problem, we have developed an approach which reveals itself useful for solving a general class of problems involving multinomial probabilities. Concerning the main problem, we provide a self-contained proof that the minimum of the multinomial lower tail is reached, as conjectured by Gupta and Nagel (Sankhya Ser. B 29 (1967) 1) (within the framework of subset-selection problems) at the equal probability configuration, i.e., when the cell probabilities are equal to one another. We also point out some novel inequalities and general properties involving multinomial probabilities and multinomial coefficients. [Copyright &y& Elsevier]

Cressie and Read (J. Roy. Statist. Soc. B 46 (1984) 440–464) introduced the power divergence statistics, Ra, as multinomial goodness-of-fit statistics. Each Ra has a limiting noncentral chi-square distribution under a local alternative and has a limiting normal distribution under a nonlocal alternative. Taneichi et al. (J. Multivariate Anal. 81 (2002) 335–359) derived an asymptotic approximation for the distribution of Ra under local alternatives. In this paper, using multivariate Edgeworth expansion for a continuous distribution, we show how the approximation based on the limiting normal distribution of Ra under nonlocal alternatives can be improved. We apply the expansion to the power approximation for Ra. The results of numerical investigation show that the proposed power approximation is very effective for the likelihood ratio test. [Copyright &y& Elsevier]

For the purpose of testing the spherical uniformity based on i.i.d. directional data (unit vectors) zi, i=1,…,n, Anderson and Stephens (Biometrika 59 (1972) 613–621) proposed testing procedures based on the statistics Smax=maxu S(u) and Smin=minu S(u), where u is a unit vector and nS(u) is the sum of squares of u′zi''s. In this paper, we also consider another test statistic Srange=Smax−Smin. We provide formulas for the P-values of Smax, Smin, Srange by approximating tail probabilities of the limiting null distributions by means of the tube method, an integral-geometric approach for evaluating tail probability of the maximum of a Gaussian random field. Monte Carlo simulations for examining the accuracy of the approximation and for the power comparison of the statistics are given. [Copyright &y& Elsevier]

Let X and Y be d-dimensional random vectors having elliptically symmetric distributions. Call X and Y affinely equivalent if Y has the same distribution as AX+b for some nonsingular d×d-matrix A and some b∈Rd. This paper studies a class of affine invariant tests for affine equivalence under certain moment restrictions. The test statistics are measures of discrepancy between the empirical distributions of the norm of suitably standardized data. [Copyright &y& Elsevier]

In this paper we propose a new test for the multivariate two-sample problem. The test statistic is the difference of the sum of all the Euclidean interpoint distances between the random variables from the two different samples and one-half of the two corresponding sums of distances of the variables within the same sample. The asymptotic null distribution of the test statistic is derived using the projection method and shown to be the limit of the bootstrap distribution. A simulation study includes the comparison of univariate and multivariate normal distributions for location and dispersion alternatives. For normal location alternatives the new test is shown to have power similar to that of the t- and T2-Test. [Copyright &y& Elsevier]

The purpose of this paper is to investigate kernel density estimators for spatial processes with linear or nonlinear structures. Sufficient conditions for such estimators to converge in L1 are obtained under extremely general, verifiable conditions. The results hold for mixing as well as for nonmixing processes. Potential applications include testing for spatial interaction, the spatial analysis of causality structures, the definition of leading/lagging sites, the construction of clusters of comoving sites, etc. [Copyright &y& Elsevier]

The classical theory for testing the null hypothesis that a set of canonical correlation coefficients is zero leads to a chi-square test under the assumption of multi-normality. The test has been used in the context of dimension reduction. In this paper, we study the limiting distribution of the test statistic without the normality assumption, and obtain a necessary and sufficient condition for the chi-square limiting distribution to hold. Implications of the result are also discussed for the problem of dimension reduction. [Copyright &y& Elsevier]

Poisson mixed models are used to analyze a wide variety of cluster count data. These models are commonly developed based on the assumption that the random effects have either the log-normal or the gamma distribution. Obtaining consistent as well as efficient estimates for the parameters involved in such Poisson mixed models has, however, proven to be difficult. Further problem gets mounted when the data are collected repeatedly from the individuals of the same cluster or family. In this paper, we introduce a generalized quasilikelihood approach to analyze the repeated familial data based on the familial structure caused by gamma random effects. This approach provides estimates of the regression parameters and the variance component of the random effects after taking the longitudinal correlations of the data into account. The estimators are consistent as well as highly efficient. [Copyright &y& Elsevier]