Generalized linear models with functional predictors

Summary. We present a technique for extending generalized linear models to the situation where some of the predictor variables are observations from a curve or function. The technique is particularly useful when only fragments of each curve have been observed. We demonstrate, on both simulated and real data sets, how this approach can be used to perform linear, logistic and censored regression with functional predictors. In addition, we show how functional principal components can be used to gain insight into the relationship between the response and functional predictors. Finally, we extend the methodology to apply generalized linear models and principal components to standard missing data problems.     

Inference in multivariate linear regression models with elliptically distributed errors

In this study we investigate the problem of estimation and testing of hypotheses in multivariate linear regression models when the errors involved are assumed to be non-normally distributed. We consider the class of heavy-tailed distributions for this purpose. Although our method is applicable for any distribution in this class, we take the multivariate t-distribution for illustration. This distribution has applications in many fields of applied research such as Economics, Business, and Finance. For estimation purpose, we use the modified maximum likelihood method in order to get the so-called modified maximum likelihood estimates that are obtained in a closed form. We show that these estimates are substantially more efficient than least-square estimates. They are also found to be robust to reasonable deviations from the assumed distribution and also many data anomalies such as the presence of outliers in the sample, etc. We further provide test statistics for testing the relevant hypothesis regarding the regression coefficients.     

Linear censored regression models with skew scale mixtures of normal distributions

A special source of difficulty in the statistical analysis is the possibility that some subjects may not have a complete observation of the response variable. Such incomplete observation of the response variable is called censoring. Censorship can occur for a variety of reasons, including limitations of measurement equipment, design of the experiment, and non-occurrence of the event of interest until the end of the study. In the presence of censoring, the dependence of the response variable on the explanatory variables can be explored through regression analysis. In this paper, we propose to examine the censorship problem in context of the class of asymmetric, i.e., we have proposed a linear regression model with censored responses based on skew scale mixtures of normal distributions. We develop a Monte Carlo EM (MCEM) algorithm to perform maximum likelihood inference of the parameters in the proposed linear censored regression models with skew scale mixtures of normal distributions. The MCEM algorithm has been discussed with an emphasis on the skew-normal, skew Student-t-normal, skew-slash and skew-contaminated normal distributions. To examine the performance of the proposed method, we present some simulation studies and analyze a real dataset.     

Likelihood-based inference for censored linear regression models with scale mixtures of skew-normal distributions

In many studies, the data collected are subject to some upper and lower detection limits. Hence, the responses are either left or right censored. A complication arises when these continuous measures present heavy tails and asymmetrical behavior; simultaneously. For such data structures, we propose a robust-censored linear model based on the scale mixtures of skew-normal (SMSN) distributions. The SMSN is an attractive class of asymmetrical heavy-tailed densities that includes the skew-normal, skew-t, skew-slash, skew-contaminated normal and the entire family of scale mixtures of normal (SMN) distributions as special cases. We propose a fast estimation procedure to obtain the maximum likelihood (ML) estimates of the parameters, using a stochastic approximation of the EM (SAEM) algorithm. This approach allows us to estimate the parameters of interest easily and quickly, obtaining as a byproducts the standard errors, predictions of unobservable values of the response and the log-likelihood function. The proposed methods are illustrated through real data applications and several simulation studies.     

Bayesian analysis of censored linear regression models with scale mixtures of normal distributions

As is the case of many studies, the data collected are limited and an exact value is recorded only if it falls within an interval range. Hence, the responses can be either left, interval or right censored. Linear (and nonlinear) regression models are routinely used to analyze these types of data and are based on normality assumptions for the errors terms. However, those analyzes might not provide robust inference when the normality assumptions are questionable. In this article, we develop a Bayesian framework for censored linear regression models by replacing the Gaussian assumptions for the random errors with scale mixtures of normal (SMN) distributions. The SMN is an attractive class of symmetric heavy-tailed densities that includes the normal, Student-t, Pearson type VII, slash and the contaminated normal distributions, as special cases. Using a Bayesian paradigm, an efficient Markov chain Monte Carlo algorithm is introduced to carry out posterior inference. A new hierarchical prior distribution is suggested for the degrees of freedom parameter in the Student-t distribution. The likelihood function is utilized to compute not only some Bayesian model selection measures but also to develop Bayesian case-deletion influence diagnostics based on the q-divergence measure. The proposed Bayesian methods are implemented in the R package BayesCR. The newly developed procedures are illustrated with applications using real and simulated data.     

Modeling nonlinear time series with local mixtures of generalized linear models

The authors consider a novel class of nonlinear time series models based on local mixtures of regressions of exponential family models, where the covariates include functions of lags of the dependent variable. They give conditions to guarantee consistency of the maximum likelihood estimator for correctly specified models, with stationary and nonstationary predictors. They show that consistency of the maximum likelihood estimator still holds under model misspecification. They also provide probabilistic results for the proposed model when the vector of predictors contains only lags of transformations of the modeled time series. They illustrate the consistency of the maximum likelihood estimator and the probabilistic properties via Monte Carlo simulations. Finally, they present an application using real data.     

Generalized symmetrical partial linear model

In this work, we propose a new model called generalized symmetrical partial linear model, based on the theory of generalized linear models and symmetrical distributions. In our model the response variable follows a symmetrical distribution such a normal, Student-t, power exponential, among others. Following the context of generalized linear models we consider replacing the traditional linear predictors by the more general predictors in whose case one covariate is related with the response variable in a non-parametric fashion, that we do not specified the parametric function. As an example, we could imagine a regression model in which the intercept term is believed to vary in time or geographical location. The backfitting algorithm is used for estimating the parameters of the proposed model. We perform a simulation study for assessing the behavior of the penalized maximum likelihood estimators. We use the quantile residuals for checking the assumption of the model. Finally, we analyzed real data set related with pH rivers in Ireland.     

Consistency of l 1 estimates in censored linear regression models

In this paper, the regression model with a nonnegativity constraint on the dependent variable is considered. Under weak conditions, L ₁ estimates of the regression coefficients are shown to be consistent.     

Generalized log-gamma regression models with cure fraction

In this paper, the generalized log-gamma regression model is modified to allow the possibility that long-term survivors may be present in the data. This modification leads to a generalized log-gamma regression model with a cure rate, encompassing, as special cases, the log-exponential, log-Weibull and log-normal regression models with a cure rate typically used to model such data. The models attempt to simultaneously estimate the effects of explanatory variables on the timing acceleration/deceleration of a given event and the surviving fraction, that is, the proportion of the population for which the event never occurs. The normal curvatures of local influence are derived under some usual perturbation schemes and two martingale-type residuals are proposed to assess departures from the generalized log-gamma error assumption as well as to detect outlying observations. Finally, a data set from the medical area is analyzed.     

Test for high dimensional regression coefficients of partially linear models

Abstract

Partially linear models attract much attention to investigate the association between predictors and the response variable when the dependency on some predictors may be nonlinear. However, the hypothesis test for significance of predictors is still challenging, especially when the number of predictors is larger than sample size. In this paper, we reconsider the test procedure of Zhong and Chen (2011 Zhong, P., and S. Chen. 2011. Tests for high-dimensional regression coefficients with factorial designs. Journal of the American Statistical Association 106 (493):260–74. doi:10.1198/jasa.2011.tm10284.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) when regression models have nonlinear components, and propose a generalized U-statistic for testing the linear components of the high dimensional partially linear models. The asymptotic properties of test statistic are obtained under null and alternative hypotheses, where the effect of nonlinear components should be considered and thus is different from that in linear models. Through simulation studies, we demonstrate good finite-sample performance of the proposed test in comparison with the existing methods. The practical utility of our proposed method is illustrated by a real data example.     

Bayesian quantile regression for skew-normal linear mixed models

Linear mixed models have been widely used to analyze repeated measures data which arise in many studies. In most applications, it is assumed that both the random effects and the within-subjects errors are normally distributed. This can be extremely restrictive, obscuring important features of within-and among-subject variations. Here, quantile regression in the Bayesian framework for the linear mixed models is described to carry out the robust inferences. We also relax the normality assumption for the random effects by using a multivariate skew-normal distribution, which includes the normal ones as a special case and provides robust estimation in the linear mixed models. For posterior inference, we propose a Gibbs sampling algorithm based on a mixture representation of the asymmetric Laplace distribution and multivariate skew-normal distribution. The procedures are demonstrated by both simulated and real data examples.     

The optimal Lp norm estimator in linear regression models

The least squares estimator is usually applied when estimating the parameters in linear regression models. As this estimator is sensitive to departures from normality in the residual distribution, several alternatives have been proposed. The L_p norm estimators is one class of such alternatives. It has been proposed that the kurtosis of the residual distribution be taken into account when a choice of estimator in the L_p norm class is made (i.e. the choice of p). In this paper, the asymtotic variance of the estimators is used as the criterion in the choice of p. It is shown that when this criterion is applied, other characteristics of the residual distribution than the kurtosis (namely moments of order p-2 and 2p-2) are important.     

Weighted composite quantile regression for partially linear varying coefficient models

Partially linear varying coefficient models (PLVCMs) with heteroscedasticity are considered in this article. Based on composite quantile regression, we develop a weighted composite quantile regression (WCQR) to estimate the non parametric varying coefficient functions and the parametric regression coefficients. The WCQR is augmented using a data-driven weighting scheme. Moreover, the asymptotic normality of proposed estimators for both the parametric and non parametric parts are studied explicitly. In addition, by comparing the asymptotic relative efficiency theoretically and numerically, WCQR method all outperforms the CQR method and some other estimate methods. To achieve sparsity with high-dimensional covariates, we develop a variable selection procedure to select significant parametric components for the PLVCM and prove the method possessing the oracle property. Both simulations and data analysis are conducted to illustrate the finite-sample performance of the proposed methods.     

Testing for a change point in linear regression models

In this paper, the Schwarz Information Criterion (SIC) is proposed to locate a change point in the simple linear regression model, as well as in the multiple linear regression model. The method is then applied to a financial data set, and a change point is successfully detected.     

A non-iterative sampling Bayesian method for linear mixed models with normal independent distributions

In this paper, we utilize normal/independent (NI) distributions as a tool for robust modeling of linear mixed models (LMM) under a Bayesian paradigm. The purpose is to develop a non-iterative sampling method to obtain i.i.d. samples approximately from the observed posterior distribution by combining the inverse Bayes formulae, sampling/importance resampling and posterior mode estimates from the expectation maximization algorithm to LMMs with NI distributions, as suggested by Tan et al. [33 Tan, M., Tian, G. and Ng, K. 2003. A noniterative sampling method for computing posteriors in the structure of EM-type algorithms. Statist. Sinica, 13(3): 625–640. [Web of Science ®] , [Google Scholar]]. The proposed algorithm provides a novel alternative to perfect sampling and eliminates the convergence problems of Markov chain Monte Carlo methods. In order to examine the robust aspects of the NI class, against outlying and influential observations, we present a Bayesian case deletion influence diagnostics based on the Kullback–Leibler divergence. Further, some discussions on model selection criteria are given. The new methodologies are exemplified through a real data set, illustrating the usefulness of the proposed methodology.     

Residual components in generalized linear models

The adequacy of a postulated generalized linear model can often be improved by transforming predictors and/or including additional explanatory variables. To assess the fit relative to a given predictor, we define its corresponding residual component. Asymptotic bias and variance of the residual component are considered, paying particular attention to the case that the presumed model is valid.     

Minimax designs for 2k factorial experiments for generalized linear models

ABSTRACT

Formulas for A- and C-optimal allocations for binary factorial experiments in the context of generalized linear models are derived. Since the optimal allocations depend on GLM weights, which often are unknown, a minimax strategy is considered. This is shown to be simple to apply to factorial experiments. Efficiency is used to evaluate the resulting design. In some cases, the minimax design equals the optimal design. For other cases no general conclusion can be drawn. An example of a two-factor logit model suggests that the minimax design performs well, and often better than a uniform allocation.