Multivariate Statistical Modelling Based on Generalized Linear Models

Journal of the Royal Statistical Society: Series C (Applied Statistics), 2016

We propose a general framework for non-normal multivariate data analysis called multivariate covariance generalized linear models (McGLMs), designed to handle multivariate response variables, along with a wide range of temporal and spatial correlation structures defined in terms of a covariance link function combined with a matrix linear predictor involving known matrices. The method is motivated by three data examples that are not easily handled by existing methods. The first example concerns multivariate count data, the second involves response variables of mixed types, combined with repeated measures and longitudinal structures, and the third involves a spatio-temporal analysis of rainfall data. The models take non-normality into account in the conventional way by means of a variance function, and the mean structure is modelled by means of a link function and a linear predictor. The models are fitted using an efficient Newton scoring algorithm based on quasi-likelihood and Pearson estimating functions, using only second-moment assumptions. This provides a unified approach to a wide variety of different types of response variables and covariance structures, including multivariate extensions of repeated measures, time series, longitudinal, spatial and spatio-temporal structures.

Interdisciplinary Information Sciences

These lecture notes provide a quick review of basic concepts in statistical analysis and probability theory for data science. We survey general description of single-and multi-variate data, and derive regression models by means of the method of least squares. As theoretical backgrounds we provide basic knowledge of probability theory which is indispensable for further study of mathematical statistics and probability models. We show that the regression line for a multi-variate normal distribution coincides with the regression curve defined through the conditional density function. In Appendix matrix operations are quickly reviewed. These notes are based on the lectures delivered in Graduate Program in Data Science (GP-DS) and Data Sciences Program (DSP) at Tohoku University in 2018-2020.

Journal of Probability and Statistics, 2013

This volume constitutes the special issue of the Recent Advances in Univariate and Multivariate Models. First, the editors wish to record their thanks to all those who helped with both the selection and referring of papers of this issue. Seventeen papers were submitted to this special issue and only five accepted in this volume represent the contributed papers selected by the editors as suitable for publication.

1987

Symmetry, 2022

The mixture of generalised linear models (MGLM) requires knowledge about each mixture component’s specific exponential family (EF) distribution. This assumption is relaxed and a mixture of semi-parametric generalised linear models (MSPGLM) approach is proposed, which allows for unknown distributions of the EF for each mixture component while much of the parametric structure of the traditional MGLM is retained. Such an approach inherently allows for both symmetric and non-symmetric component distributions, frequently leading to non-symmetrical response variable distributions. It is assumed that the random component of each mixture component follows an unknown distribution of the EF. The specific member can either be from the standard class of distributions or from the broader set of admissible distributions of the EF which is accessible through the semi-parametric procedure. Since the inverse link functions of the mixture components are unknown, the MSPGLM estimates each mixture comp...

John Wiley & Sons, Ltd eBooks, 2010

Generalized linear models provide a general framework for handling regression modeling for normal and non-normal data, including multiple linear regression, ANOVA, logistic regression, Poisson regression and log-linear models for contingency tables. All the major statistical packages include facilities for tting generalized linear models. A generalized linear model is de ned by choosing a link function and a variance function, along with choosing a response variable and a set of explanatory variables. The link function transforms the mean of the response variable to a scale where the model is linear. The variance function describes how the variance behaves as a function of the mean. Each choice of variance function corresponds to a certain deviance function, and model tting is accomplished by minimizing the deviance, generalizing least squares tting. Inference on parameters, and hypothesis testing is performed by means of analysis of deviance, generalization the classical ANOVA method. Estimation and analysis of deviance are based on quasi-likelihood methods, 1 requiring only second-moment assumptions, thereby providing a certain robustness against misspeci cation of the probability model. The choice of link and variance functions may be checked by means of residual analysis.

Servicio Editorial de la Universidad del País Vasco/Euskal Herriko Unibertsitatearen Argitalpen Zerbitzua eBooks, 2020

2012

This paper address the problem of estimating the parameter of a Multivariate Generalized Gaussian Distribution (MGGD). After a brief introduction of the MGG distribution, the maximum likelihood estimators of the MGGD parameters are given. For β ∈]0, 1], which corresponds to most of the real-life problems, we prove theoretically that the maximum likelihood estimator of the normalized covariance matrix exists and is unique up to a scalar factor. Some experiments are conducted to evaluate the convergence speed of the proposed estimation algorithm. With simulations results, we show that the maximum likelihood estimator of the normalized covariance matrix is unbiased and consistent. Concerning the shape parameter β, we show experimentally that the variance of its maximum likelihood estimator reaches the Cramér-Rao lower bound.

1983

A multidimensional extension of the two-parameter logistic latent trait model is presented and some of its characteristics are discussed. In addition, sufficient statistics for the parameters of the model are derived, as -is the information function. Finally,the estimation of the parameters of the model using the maximum likelihood estimation technique is also discussed.

Mathematics, 2022

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