Statistical inference in partially-varying-coefficient single-index model

Consider a varying-coefficient single-index model which consists of two parts: the linear part with varying coefficients and the nonlinear part with a single-index structure, and are hence termed as varying-coefficient single-index models. This model includes many important regression models such as single-index models, partially linear single-index models, varying-coefficient model and varying-coefficient partially linear models as special examples. In this paper, we mainly study estimating problems of the varying-coefficient vector, the nonparametric link function and the unknown parametric vector describing the single-index in the model. A stepwise approach is developed to obtain asymptotic normality estimators of the varying-coefficient vector and the parametric vector, and estimators of the nonparametric link function with a convergence rate. The consistent estimator of the structural error variance is also obtained. In addition, asymptotic pointwise confidence intervals and confidence regions are constructed for the varying coefficients and the parametric vector. The bandwidth selection problem is also considered. A simulation study is conducted to evaluate the proposed methods, and real data analysis is also used to illustrate our methods.     

Marginal permutation invariant covariance matrices with applications to linear models

The goal of the present paper is to perform a comprehensive study of the covariance structures in balanced linear models containing random factors which are invariant with respect to marginal permutations of the random factors. We shall focus on model formulation and interpretation rather than the estimation of parameters. It is proven that permutation invariance implies a specific structure for the covariance matrices. Useful results are obtained for the spectra of permutation invariant covariance matrices. In particular, the reparameterization of random effects, i.e., imposing certain constraints, will be considered. There are many possibilities to choose reparameterization constraints in a linear model, however not every reparameterization keeps permutation invariance. The question is if there are natural restrictions on the random effects in a given model, i.e., such reparameterizations which are defined by the covariance structure of the corresponding factor. Examining relationships between the reparameterization conditions applied to the random factors of the models and the spectrum of the corresponding covariance matrices when permutation invariance is assumed, restrictions on the spectrum of the covariance matrix are obtained which lead to “sum-to-zero” reparameterization of the corresponding factor.     

Local influence analysis of multivariate probit latent variable models

The multivariate probit model is very useful for analyzing correlated multivariate dichotomous data. Recently, this model has been generalized with a confirmatory factor analysis structure for accommodating more general covariance structure, and it is called the MPCFA model. The main purpose of this paper is to consider local influence analysis, which is a well-recognized important step of data analysis beyond the maximum likelihood estimation, of the MPCFA model. As the observed-data likelihood associated with the MPCFA model is intractable, the famous Cook's approach cannot be applied to achieve local influence measures. Hence, the local influence measures are developed via Zhu and Lee's [Local influence for incomplete data model, J. Roy. Statist. Soc. Ser. B 63 (2001) 111-126.] approach that is closely related to the EM algorithm. The diagnostic measures are derived from the conformal normal curvature of an appropriate function. The building blocks are computed via a sufficiently large random sample of the latent response strengths and latent variables that are generated by the Gibbs sampler. Some useful perturbation schemes are discussed. Results that are obtained from analyses of an artificial example and a real example are presented to illustrate the newly developed methodology.     

Restricted expected multivariate least squares

A new approach of estimating parameters in multivariate models is introduced. A fitting function will be used. The idea is to estimate parameters so that the fitting function equals or will be close to its expected value. The function will be decomposed into two parts. From one part, which will be independent of the mean parameters, the dispersion matrix is estimated. This estimator is inserted in the second part which then yields the estimators of the mean parameters. The Growth Curve model, extended Growth Curve model and a multivariate variance components model will illustrate the approach.     

Positivity for Gaussian graphical models

Gaussian graphical models are parametric statistical models for jointly normal random variables whose dependence structure is determined by a graph. In previous work, we introduced trek separation, which gives a necessary and sufficient condition in terms of the graph for when a subdeterminant is zero for all covariance matrices that belong to the Gaussian graphical model. Here we extend this result to give explicit cancellation-free formulas for the expansions of non-zero subdeterminants.     

On testing equality of pairwise rank correlations in a multivariate random vector

Spearman’s rank-correlation coefficient (also called Spearman’s rho) represents one of the best-known measures to quantify the degree of dependence between two random variables. As a copula-based dependence measure, it is invariant with respect to the distribution’s univariate marginal distribution functions. In this paper, we consider statistical tests for the hypothesis that all pairwise Spearman’s rank correlation coefficients in a multivariate random vector are equal. The tests are nonparametric and their asymptotic distributions are derived based on the asymptotic behavior of the empirical copula process. Only weak assumptions on the distribution function, such as continuity of the marginal distributions and continuous partial differentiability of the copula, are required for obtaining the results. A nonparametric bootstrap method is suggested for either estimating unknown parameters of the test statistics or for determining the associated critical values. We present a simulation study in order to investigate the power of the proposed tests. The results are compared to a classical parametric test for equal pairwise Pearson’s correlation coefficients in a multivariate random vector. The general setting also allows the derivation of a test for stochastic independence based on Spearman’s rho.     

Nonparametric likelihood ratio goodness-of-fit tests for survival data

Berk and Jones (Z. Wahrsch. Verw. Gebiete 47 (1979) 47) described a nonparametric likelihood test of uniformity that is more efficient, in Bahadur's sense, than any weighted Kolmogorov-Smirnov test at any alternative. This article shows how to obtain a nonparametric likelihood test of a general parametric family for incomplete survival data. A nonparametric likelihood ratio test process is employed to measure the discrepancy between a parametric family and the observed data. Large sample properties of the likelihood ratio test process are studied under both the null and alternative hypotheses. A Monte Carlo simulation method is proposed to estimate its null distribution. We show how to produce a likelihood ratio graphical check as well as a formal test of a parametric family based on the developed theory. Our method is developed for the right-censorship model, but can be easily extended to some other survival models. Illustrations are given using both real and simulated data.     

Conditional orderings and positive dependence

Every univariate random variable is smaller, with respect to the ordinary stochastic order and with respect to the hazard rate order, than a right censored version of it. In this paper we attempt to generalize these facts to the multivariate setting. It turns out that in general such comparisons do not hold in the multivariate case, but they do under some assumptions of positive dependence. First we obtain results that compare the underlying random vectors with respect to the usual multivariate stochastic order. A larger slew of results, that yield comparisons of the underlying random vectors with respect to various multivariate hazard rate orders, is given next. Some comparisons with respect to the orthant orders are also discussed.     

Uncertainty under a multivariate nested-error regression model with logarithmic transformation

This work aims to predict exponentials of mixed effects under a multivariate linear regression model with one random factor. Such quantities are of particular interest in prediction problems where the dependent variable is the logarithm of the variable that is the object of inference. Bias-corrected empirical predictors of the target quantities are defined. A second-order approximation for the mean crossed product error of two of these predictors is obtained, where the mean squared error is a particular case. An estimator of the mean crossed product error with second-order bias is proposed. Finally, results are illustrated through an application related to small area estimation.     

Model testing for multivariate censored data

Summary. In the fields like Astronomy and Ecology, the need for proper statistical analysis of data that are censored is being increasingly recognized. Such data occur when, due to noise or other factors, instruments fail to detect low luminosities of celestial objects, or low concentrations of certain pollutants. For multivariate censored data sets there are very few distribution free methods available and researchers in the various fields often impose an assumption on the joint distribution, such as multivariate normality, and carry out parametric inferences. Under censoring, however, such parametric inferences are asymptotically wrong if the imposed assumption is incorrect. In this paper we propose a class of goodness-of-fit procedures for testing assumptions about the multivariate distribution under random censoring. The test procedures generalize Pearson's goodness-of-fit test in the sense that they are based on the concept of observed-minus-expected frequencies. The theory of the test statistic, however, differs from that for the classical Pearson test due to the accommodation of censored data. Received: 24 May 1994 / In revised form: 3 March 1996     

An extension of Fisher's discriminant analysis for stochastic processes

In this paper we present a general notion of Fisher's linear discriminant analysis that extends the classical multivariate concept to situations that allow for function-valued random elements. The development uses a bijective mapping that connects a second order process to the reproducing kernel Hilbert space generated by its within class covariance kernel. This approach provides a seamless transition between Fisher's original development and infinite dimensional settings that lends itself well to computation via smoothing and regularization. Simulation results and real data examples are provided to illustrate the methodology.     

Construction of asymmetric multivariate copulas

In this paper we introduce two methods for the construction of asymmetric multivariate copulas. The first is connected with products of copulas. The second approach generalises the Archimedean copulas. The resulting copulas are asymmetric and may have more than two parameters in contrast to most of the parametric families of copulas described in the literature. We study the properties of the proposed families of copulas such as the dependence of two components (Kendall’s tau, tail dependence), marginal distributions and the generation of random variates.     

Characterization of dependence of multidimensional Lévy processes using Lévy copulas

This paper suggests Lévy copulas in order to characterize the dependence among components of multidimensional Lévy processes. This concept parallels the notion of a copula on the level of Lévy measures. As for random vectors, a version of Sklar's theorem states that the law of a general multivariate Lévy process is obtained by combining arbitrary univariate Lévy processes with an arbitrary Lévy copula. We construct parametric families of Lévy copulas and prove a limit theorem, which indicates how to obtain the Lévy copula of a multivariate Lévy process X from the ordinary copula of the random vector X_t for small t.     

On Hadamard differentiability in k-sample semiparametric models—with applications to the assessment of structural relationships

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.     

Restricted estimation in multivariate measurement error regression model

We study a multivariate ultrastructural measurement error (MUME) model with more than one response variable. This model is a synthesis of multivariate functional and structural models. Three consistent estimators of regression coefficients, satisfying the exact linear restrictions have been proposed. Their asymptotic distributions are derived under the assumption of a non-normal measurement error and random error components. A simulation study is carried out to investigate the small sample properties of the estimators. The effect of departure from normality of the measurement errors on the estimators is assessed.     

Bayesian reference analysis for Gaussian Markov random fields

Gaussian Markov random fields (GMRF) are important families of distributions for the modeling of spatial data and have been extensively used in different areas of spatial statistics such as disease mapping, image analysis and remote sensing. GMRFs have been used for the modeling of spatial data, both as models for the sampling distribution of the observed data and as models for the prior of latent processes/random effects; we consider mainly the former use of GMRFs. We study a large class of GMRF models that includes several models previously proposed in the literature. An objective Bayesian analysis is presented for the parameters of the above class of GMRFs, where explicit expressions for the Jeffreys (two versions) and reference priors are derived, and for each of these priors results on posterior propriety of the model parameters are established. We describe a simple MCMC algorithm for sampling from the posterior distribution of the model parameters, and study frequentist properties of the Bayesian inferences resulting from the use of these automatic priors. Finally, we illustrate the use of the proposed GMRF model and reference prior for studying the spatial variability of lip cancer cases in the districts of Scotland over the period 1975-1980.     

On Mardia's and Song's measures of kurtosis in elliptical distributions

The main objective of this paper is the calculation and the comparative study of two general measures of multivariate kurtosis, namely Mardia's measure β_2,p and Song's measure S(f). In this context, general formulas for the said measures are derived for the broad family of the elliptically contoured symmetric distributions and also for specific members of this family, like the multivariate t-distribution, the multivariate Pearson type II, the multivariate Pearson type VII, the multivariate symmetric Kotz type distribution and the uniform distribution in the unit sphere. Analytic expressions for computing Shannon and Rényi entropies are obtained under the elliptic family. The behaviour of Mardia's and Song's measures, their similarities and differences, possible interpretations and uses in practice are investigated by comparing them in specific members of the elliptic family of multivariate distributions. An empirical estimator of Song's measure is moreover proposed and its asymptotic distribution is investigated under the elliptic family of multivariate distributions.     

Nonparametric methods for unbalanced multivariate data and many factor levels

We propose different nonparametric tests for multivariate data and derive their asymptotic distribution for unbalanced designs in which the number of factor levels tends to infinity (large a, small n_i case). Quasi gratis, some new parametric multivariate tests suitable for the large a asymptotic case are also obtained. Finite sample performances are investigated and compared in a simulation study. The nonparametric tests are based on separate rankings for the different variables. In the presence of outliers, the proposed nonparametric methods have better power than their parametric counterparts. Application of the new tests is demonstrated using data from plant pathology.     

Case deletion diagnostics in multilevel models

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.     

Multivariate extreme value copulas with factor and tree dependence structures

Parsimonious extreme value copula models with O(d) parameters for d observed variables of extrema are presented. These models utilize the dependence characteristics, including factor and tree structures, assumed on the underlying variables that give rise to the data of extremes. For factor structures, a class of parametric models is obtained by taking the extreme value limit of factor copulas with non-zero tail dependence. An alternative model suitable for both factor and tree structures imposes constraints on the parametric Hüsler-Reiss copula to get representations in terms of O(d) other parameters. Dependence properties are discussed. As the full density is often intractable, the method of composite (pairwise) likelihood is used for model inference. Procedures to improve the stability of bivariate density evaluation are also developed. The proposed models are applied to two data examples — one for annual extreme river flows and one for bimonthly extremes of daily stock returns.