Contributions To Multivariate Analysis Based On Elliptic T Model

Theoretical developments in multivariate analysis are primarily based on the assumption of multivariate normality and very little is known for other cases. The aim of the present work is to generalize results of multivariate analysis based on a class of elliptic distributions, more specifically the subclass of the multivariate t-distributions with suitable parameters, rather than the usual normality assumption. The multivariate normal distribution belongs to the class of elliptic as well as the subclass of t-distribution.;The major contributions of the thesis are: (a) An elliptic set-up for uncorrelated samples is proposed. (b) The distributions of sample mean and covariance matrix are derived. (c) Classification problem is studied for the elliptic set-up.;The elliptic class is further specialized to a subclass of multivariate t-distributions, whose characteristic function, conditional distributions etc. are derived. Also the above problems (b) and (c) are studied. In addition the following problems have been solved: (i) null and non-null distributions of quadratic forms (analogue of non-central chi-square). (ii) estimation of location, scale and degrees of freedom parameters of the t-distribution and the sampling properties of the estimators. (iii) orthogonal factor analysis when both observed error and unobserved factors follow multivariate t-distributions. (iv) estimation of parameters and testing of hypothesis for a regression model with error variable having a multivariate t-distribution

2015 International Symposium in Statistics

This proceedings volume contains eight selected papers that were presented in the International Symposium in Statistics (ISS) 2015 On Advances in Parametric and Semi-parametric Analysis of Multivariate, Time Series, Spatial-temporal, and Familial-longitudinal Data, held in St. John’s, Canada from July 6 to 8, 2015. The main objective of the ISS-2015 was the discussion on advances and challenges in parametric and semi-parametric analysis for correlated data in both continuous and discrete setups. Thus, as a reflection of the theme of the symposium, the eight papers of this proceedings volume are presented in four parts. Part I is comprised of papers examining Elliptical t Distribution Theory. In Part II, the papers cover spatial and temporal data analysis. Part III is focused on longitudinal multinomial models in parametric and semi-parametric setups. Finally Part IV concludes with a paper on the inferences for longitudinal data subject to a challenge of important covariates selection from a set of large number of covariates available for the individuals in the study

Dynamic Mixed Models for Familial Longitudinal Data

This book provides a theoretical foundation for the analysis of discrete data such as count and binary data in the longitudinal setup. Unlike the existing books, this book uses a class of auto-correlation structures to model the longitudinal correlations for the repeated discrete data that accommodates all possible Gaussian type auto-correlation models as special cases including the equi-correlation models. This new dynamic modelling approach is utilized to develop theoretically sound inference techniques such as the generalized quasi-likelihood (GQL) technique for consistent and efficient es

Longitudinal categorical data analysis

This is the first book in longitudinal categorical data analysis with parametric correlation models developed based on dynamic relationships among repeated categorical responses. This book is a natural generalization of the longitudinal binary data analysis to the multinomial data setup with more than two categories. Thus, unlike the existing books on cross-sectional categorical data analysis using log linear models, this book uses multinomial probability models both in cross-sectional and longitudinal setups. A theoretical foundation is provided for the analysis of univariate multinomial responses, by developing models systematically for the cases with no covariates as well as categorical covariates, both in cross-sectional and longitudinal setups. In the longitudinal setup, both stationary and non-stationary covariates are considered. These models have also been extended to the bivariate multinomial setup along with suitable covariates. For the inferences, the book uses the generalized quasi-likelihood as well as the exact likelihood approaches. The book is technically rigorous, and, it also presents illustrations of the statistical analysis of various real life data involving univariate multinomial responses both in cross-sectional and longitudinal setups. This book is written mainly for the graduate students and researchers in statistics and social sciences, among other applied statistics research areas. However, the rest of the book, specifically the chapters from 1 to 3, may also be used for a senior undergraduate course in statistics. Brajendra Sutradhar is a University Research Professor at Memorial University in St. John's, Canada. He is author of the book Dynamic Mixed Models for Familial Longitudinal Data, published in 2011 by Springer, New York. Also, he edited the special issue of the Canadian Journal of Statistics (2010, Vol. 38, June Issue, John Wiley) and the Lecture Notes in Statistics (2013, Vol. 211, Springer), with selected papers from two symposiums: ISS-2009 and ISS-2012, respectively

A multivariate approach for estimating the random effects variance component in one-way random effects model

It is well known that the ANOVA estimator of the random effects variance component in one-way random effects model can assume negative values. It is also well known that nonnegative quadratic unbiased estimators do not exist for estimating the random effects variance component (LaMotte, 1973). LaMotte (1985) indicated the possibility that nonnegative invariant quadratic estimator of the random effects variance component uniformly better than the ANOVA estimator may exist for the balanced one-way random effects model. Mathew et al. (1992a) have shown that such estimator exists only when the number of treatments is 9 or less. As noted in Herbach (1959) (see also Thompson, 1962), a simple truncation of the ANOVA estimator at zero yields uniform improvement over the ANOVA estimator. The estimators suggested by Herbach and Thompson are, in fact, restricted maximum likelihood estimators, and they are nonquadratic by nature. In this paper, we discuss a multivariate technique which always yields positive estimate of the random effects variance component in one-way random effects model. The multivariate approach exploits the estimates of the eigenvalues of the covariance matrix of the model in estimating the variance components including the error variance. The resulting estimates are nonquadratic. The success of this multivariate approach depends on the precise estimation of the eigenvalues. Since there does not exist any unbiased estimation procedure in the small sample case for the estimation of the eigenvalues, we use a delete-d jackknife procedure to estimate them. This delete-d jackknife based multivariate approach yields better estimates (in terms of mean squared error) for the random effects variance component than the restricted maximum likelihood estimation as well as Chow and Shao's (1988) nonquadratic estimation approaches, which is shown through a simulation study for the cases with number of treatments up to 20.Covariance matrix of the model Eigenvalues Positive estimates of variance components Delete jackknife Restricted maximum likelihood estimates Monte-Carlo experiment

Semi-parametric Dynamic Models for Longitudinal Ordinal Categorical Data

The over all regression function in a semi-parametric model involves a partly specified regression function in some primary covariates and a non-parametric function in some other secondary covariates. This type of semi-parametric models in a longitudinal setup has recently been discussed extensively both for repeated Poisson and negative binomial count data. However, when it is appropriate to interpret the longitudinal binary responses through a binary dynamic logits model, the inferences for semi-parametric Poisson and negative binomial models cannot be applied to such binary models as these models unlike the count data models produce recursive means and variances containing the dynamic dependence or correlation parameters. In this paper, we consider a general multinomial dynamic logits model in a semi-parametric setup first to analyze nominal categorical data in a semi-parametric longitudinal setup, and then modify this model to analyze ordinal categorical data. The ordinal responses are fitted by using a cumulative semi-parametric multinomial dynamic logits model. For the benefits of practitioners, a step by step estimation approach is developed for the non-parametric function, and for both regression and dynamic dependence parameters. In summary, a kernel-based semi-parametric weighted likelihood approach is used for the estimation of the non-parametric function. This weighted likelihood estimate for the non-parametric function is shown to be consistent. The regression and dynamic dependence parameters of the model are estimated by maximizing an approximate semi-parametric likelihood function for the parameters, which is constructed by replacing the non-parametric function with its consistent estimate. Asymptotic properties including the proofs for the consistency of the likelihood estimators of the regression and dynamic dependence parameters are discussed

A parameter dimension-split based asymptotic regression estimation theory for a multinomial panel data model

In this paper we revisit the so-called non-stationary regression models for repeated categorical/multinomial data collected from a large number of independent individuals. The main objective of the study is to obtain consistent and efficient regression estimates after taking the correlations of the repeated multinomial data into account. The existing (1) ‘working’ odds ratios based GEE (generalized estimating equations) approach has both consistency and efficiency drawbacks. Specifically, the GEE-based regression estimates can be inconsistent which is a serious limitation. Some other existing studies use a MDL (multinomial dynamic logits) model among the repeated responses. As far as the estimation of the regression effects and dynamic dependence (i.e., correlation) parameters is concerned, they use either (2) a marginal or (3) a joint likelihood approach. In the marginal approach, the regression parameters are estimated for known correlation parameters by solving their respective marginal likelihood estimating equations, and similarly the correlation parameters are estimated by solving their likelihood equations for known regression estimates. Thus, this marginal approach is an iterative approach which may not provide quick convergence. In the joint likelihood approach, the regression and correlation parameters are estimated simultaneously by searching the maximum value of the likelihood function with regard to these parameters together. This approach may encounter computational drawback, specially when the number of correlation parameters gets large. In this paper, we propose a new estimation approach where the likelihood function for the regression parameters is developed from the joint likelihood function by replacing the correlation parameter with a consistent estimator involving unknown regression parameters. Thus the new approach relaxes the dimension issue, that is, the dimension of the correlation parameters does not affect the estimation of the main regression parameters. The asymptotic properties of the estimates of the main regression parameters (obtained based on consistent estimating functions for correlation parameters) are studied in detail