Variable Selection By Regularization Methods For Generalized Mixed Models
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Variable Selection by Regularization Methods for Generalized Mixed Models
A regression analysis describes the dependency of random variables in the form of a functional relationship. One distinguishes between the dependent response variable and one or more independent influence variables. There is a variety of model classes and inference methods available, ranging from the conventional linear regression model up to recent non- and semiparametric regression models. The so-called generalized regression models form a methodically consistent framework incorporating many regression approaches with response variables that are not necessarily normally distributed, including the conventional linear regression model based on the normal distribution assumption as a special case. When repeated measurements are modeled in addition to fixed effects also random effects or coefficients can be included. Such models are known as Random Effects Models or Mixed Models. As a consequence, regression procedures are applicable extremely versatile and consider very different problems. In this dissertation regularization techniques for generalized mixed models are developed that are able to perform variable selection. These techniques are especially appropriate when many potential influence variables are present and existing approaches tend to fail. First of all a componentwise boosting technique for generalized linear mixed models is presented which is based on the likelihood function and works by iteratively fitting the residuals using weak learners. The complexity of the resulting estimator is determined by information criteria. For the estimation of variance components two approaches are considered, an estimator resulting from maximizing the profile likelihood, and an estimator which can be calculated using an approximative EM-algorithm. Then the boosting concept is extended to mixed models with ordinal response variables. Two different types of ordered models are considered, the threshold model, also known as cumulative model, and the sequential model. Both are based on the assumption that the observed response variable results from a categorized version of a latent metric variable. In the further course of the thesis the boosting approach is extended to additive predictors. The unknown functions to be estimated are expanded in B-spline basis functions, whose smoothness is controlled by penalty terms. Finally, a suitable L1-regularization technique for generalized linear models is presented, which is based on a combination of Fisher scoring and gradient optimization. Extensive simulation studies and numerous applications illustrate the competitiveness of the methods constructed in this thesis compared to conventional approaches. For the calculation of standard errors bootstrap methods are used.
Multivariate Statistical Modelling Based on Generalized Linear Models
Author: Ludwig Fahrmeir
language: en
Publisher: Springer Science & Business Media
Release Date: 2013-11-11
Classical statistical models for regression, time series and longitudinal data provide well-established tools for approximately normally distributed vari ables. Enhanced by the availability of software packages these models dom inated the field of applications for a long time. With the introduction of generalized linear models (GLM) a much more flexible instrument for sta tistical modelling has been created. The broad class of GLM's includes some of the classicallinear models as special cases but is particularly suited for categorical discrete or nonnegative responses. The last decade has seen various extensions of GLM's: multivariate and multicategorical models have been considered, longitudinal data analysis has been developed in this setting, random effects and nonparametric pre dictors have been included. These extended methods have grown around generalized linear models but often are no longer GLM's in the original sense. The aim of this book is to bring together and review a large part of these recent advances in statistical modelling. Although the continuous case is sketched sometimes, thoughout the book the focus is on categorical data. The book deals with regression analysis in a wider sense including not only cross-sectional analysis but also time series and longitudinal data situations. We do not consider problems of symmetrical nature, like the investigation of the association structure in a given set of variables. For example, log-linear models for contingency tables, which can be treated as special cases of GLM's are totally omitted. The estimation approach that is primarily considered in this book is likelihood-based.
Effective Statistical Learning Methods for Actuaries I
This book summarizes the state of the art in generalized linear models (GLMs) and their various extensions: GAMs, mixed models and credibility, and some nonlinear variants (GNMs). In order to deal with tail events, analytical tools from Extreme Value Theory are presented. Going beyond mean modeling, it considers volatility modeling (double GLMs) and the general modeling of location, scale and shape parameters (GAMLSS). Actuaries need these advanced analytical tools to turn the massive data sets now at their disposal into opportunities. The exposition alternates between methodological aspects and case studies, providing numerical illustrations using the R statistical software. The technical prerequisites are kept at a reasonable level in order to reach a broad readership. This is the first of three volumes entitled Effective Statistical Learning Methods for Actuaries. Written by actuaries for actuaries, this series offers a comprehensive overview of insurance data analytics with applications to P&C, life and health insurance. Although closely related to the other two volumes, this volume can be read independently.