Lack-of-Fit Tests for Generalized Linear Models via Splines

Cubic B-splines are used to estimate the nonparametric component of a semiparametric generalized linear model. A penalized log-likelihood ratio test statistic is constructed for the null hypothesis of the linearity of the nonparametric function. When the number of knots is fixed, its limiting null distribution is the distribution of a linear combination of independent chi-squared random variables, each with one df. The smoothing parameter is determined by giving a specified value for its asymptotically expected value under the null hypothesis. A simulation study is conducted to evaluate its power performance; a real-life dataset is used to illustrate its practical use.     

Second-Order Covariance Matrix Formula for Heteroskedastic Generalized Linear Models

For the first time, we provide a matrix formula for second-order covariances of maximum likelihood estimates in heteroskedastic generalized linear models, thus generalizing the results of Cordeiro (2004 Cordeiro , G. M. ( 2004 ). Second-order covariance matrix of maximum likelihood estimates in generalized linear models . Statist. Probab. Lett. 66 : 153 – 160 .[Crossref], [Web of Science ®] , [Google Scholar]) and Cordeiro et al. (2006 Cordeiro , G. M. , Barroso , L. P. , Botter , D. A. (2006). Covariance matrix formula for generalized linear models with unknown dispersion. Commun. Statist. Theor. Meth. 35:113–120.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) related to the generalized linear models with known and unknown dispersion parameter, respectively. The covariance matrix formula does not involve cumulants of log-likelihood derivatives and can be easily obtained using simple matrix operations. We apply our main result to a simple model. Some simulations show that the second-order covariances can be quite pronounced in small to moderate samples. The usual covariances of the maximum likelihood estimates can be corrected by these second-order covariances.     

Generalized Empirical Likelihood Inference in Generalized Linear Models for Longitudinal Data

In this article, the generalized linear model for longitudinal data is studied. A generalized empirical likelihood method is proposed by combining generalized estimating equations and quadratic inference functions based on the working correlation matrix. It is proved that the proposed generalized empirical likelihood ratios are asymptotically chi-squared under some suitable conditions, and hence it can be used to construct the confidence regions of the parameters. In addition, the maximum empirical likelihood estimates of parameters are obtained, and their asymptotic normalities are proved. Some simulations are undertaken to compare the generalized empirical likelihood and normal approximation-based method in terms of coverage accuracies and average areas/lengths of confidence regions/intervals. An example of a real data is used for illustrating our methods.     

Hierarchical Generalized Linear Models and Frailty Models with Bayesian Nonparametric Mixing

This paper proposes Bayesian nonparametric mixing for some well-known and popular models. The distribution of the observations is assumed to contain an unknown mixed effects term which includes a fixed effects term, a function of the observed covariates, and an additive or multiplicative random effects term. Typically these random effects are assumed to be independent of the observed covariates and independent and identically distributed from a distribution from some known parametric family. This assumption may be suspect if either there is interaction between observed covariates and unobserved covariates or the fixed effects predictor of observed covariates is misspecified. Another cause for concern might be simply that the covariates affect more than just the location of the mixed effects distribution. As a consequence the distribution of the random effects could be highly irregular in modality and skewness leaving parametric families unable to model the distribution adequately. This paper therefore proposes a Bayesian nonparametric prior for the random effects to capture possible deviances in modality and skewness and to explore the observed covariates' effect on the distribution of the mixed effects.     

Likelihood Inference for Spatial Generalized Linear Mixed Models

Spatial modeling is widely used in environmental sciences, biology, and epidemiology. Generalized linear mixed models are employed to account for spatial variations of point-referenced data called spatial generalized linear mixed models (SGLMMs). Frequentist analysis of these type of data is computationally difficult. On the other hand, the advent of the Markov chain Monte Carlo algorithm has made the Bayesian analysis of SGLMM computationally convenient. Recent introduction of the method of data cloning, which leads to maximum likelihood estimate, has made frequentist analysis of mixed models also equally computationally convenient. Recently, the data cloning was employed to estimate model parameters in SGLMMs, however, the prediction of spatial random effects and kriging are also very important. In this article, we propose a frequentist approach based on data cloning to predict (and provide prediction intervals) spatial random effects and kriging. We illustrate this approach using a real dataset and also by a simulation study.     

D-optimal Factorial Designs under Generalized Linear Models

Generalized linear models (GLMs) have been used widely for modeling the mean response both for discrete and continuous random variables with an emphasis on categorical response. Recently Yang, Mandal and Majumdar (2013 Yang, J., Mandal, A., Majumdar, D. (2013). Optimal designs for 2^k factorial experiments with binary response. Technical Report, Available at: http://arxiv.org/pdf/1109.5320v4.pdf. [Google Scholar]) considered full factorial and fractional factorial locally D-optimal designs for binary response and two-level experimental factors. In this article, we extend their results to a general setup with response belonging to a single-parameter exponential family and for multilevel predictors.     

Robust Inference in Generalized Linear Models

Robust inference on the parameters in generalized linear models is performed using the weighted likelihood method. Two cases are considered: a case with replicated observations and a case with a single observation of the dependent variable for each combination of the explanatory variables. The first case is common in the design of experiments, while the second case arises in observational studies. Theoretical and computational results on real datasets are presented and compared with other existing techniques.     

Covariance Matrix Formula for Generalized Linear Models with Unknown Dispersion

In this paper, the existence of the Uniformly Minimum Risk Equivariant (UMRE) estimator of parameters in SURE model under some quadratic losses and matrix losses is studied. The necessary and sufficient conditions for existence of the UMRE estimator of linearly estimable function vectors of regression coefficients under an affine group of transformations are obtained. It is proved that no UMRE estimator of the covariance matrix under any one of two affine groups of transformations exists.     

Empirical Likelihood for Generalized Partially Linear Single-index Models

Empirical-likelihood based inference for the parameters in a generalized partially linear single-index models (GPLSIM) is investigated. Based on the local linear estimators of the nonparametric parts of the GPLSIM, an estimated empirical likelihood-based statistic of the parametric components is proposed. We show that the resulting statistic is asymptotically standard chi-squared distributed, the confidence regions for the parametric components are constructed. Some simulations are conducted to illustrate the proposed method.     

Generalized Inference for a Class of Linear Models Under Heteroscedasticity

In this article we study inferences for a class of linear models under heteroscedasticity. Using the generalized inference approach, we obtain the generalized p-values of two-sided hypotheses for the multi-dimensional location parameters and one-sided hypotheses for the scale parameters, respectively. Some frequentist properties in small-sample cases and large-sample cases are proven.     

The CUSUM Test for Detecting Structural Changes in Strong Mixing Processes

Strong mixing property holds for a broad class of linear and nonlinear time series models such as Auto-Regressive Moving Average Processes and Generalized Auto-Regressive Conditional Heteroscedasticity Processes models. In this article, we study correlation structure of strong mixing sequences, and some asymptotic properties are presented. We also present a new method for detecting change point in correlation structure of strong mixing sequences, and present a nonparametric sequential analysis for detecting changes named cumulative sum test statistic for this. Asymptotic consistency of this test statistics is shown. This method is applied to simulated data of some linear and nonlinear models and power of the test is evaluated. For linear models, it is shown that this method has a better performance in comparison to Berkes et al. (2009 Berkes, I., Gombay, E., Horvath, L. (2009). Testing for changes in the covariance structure of linear processes. J. Stat. Plan. Inf. 139:2044–2063.[Crossref], [Web of Science ®] , [Google Scholar]).     

联合广义线性模型中的变量选择

在联合广义线性模型中,散度参数与均值都被赋予了广义线性模型的结构,本文主要考虑在只有分布的一阶矩和二阶矩指定的条件下,联合广义线性模型中均值部分的变量选择问题。本文采用广义拟似然函数,提出了新的模型选择准则(EAIC);该准则是Akaike信息准则的推广。论文通过模拟研究验证了该准则的效果。     

Estimation in Partially Linear Single-Index Models with Missing Covariates

In this article, we consider a partially linear single-index model Y = g(Z ^τθ₀) + X ^τβ₀ + ? when the covariate X may be missing at random. We propose weighted estimators for the unknown parametric and nonparametric part by applying weighted estimating equations. We establish normality of the estimators of the parameters and asymptotic expansion for the estimator of the nonparametric part when the selection probabilities are unknown. Simulation studies are also conducted to illustrate the finite sample properties of these estimators.     

Asymptotic Properties of Quasi-Maximum Likelihood Estimates in Generalized Linear Models

This article introduces a novel method, named JC ₁, for obtaining G-efficient mixture design to fit quadratic models. The advantage of JC ₁ method over existing algorithms is that it gives G-efficient designs without need of generating all the extreme vertices, edge centroids and constraint plane centroids of the mixture experimental region. The performance of the new method is illustrated and its comparison is given with popularly used algorithms—Snee (1975) algorithm and Welch (1985 Welch , W. J. ( 1985 ). ACED: Algorithms for the construction of experimental designs . Amer. Statistician 39 : 146 .[Crossref] , [Google Scholar]) ACED algorithm for second-order (quadratic model) designs and it is observed that JC ₁ method performs as well as the existing methods or sometimes better than those with additional advantage of large savings in computational efforts.     

Model Checks for Generalized Linear Models

In this paper we propose and study non-parametric tests for the validity of (composite) Generalized Linear Models with a given parametric link structure, which are based on certain empirical processes marked by the residuals. When properly transformed to their innovation part the resulting test statistics are distribution-free. The method perfectly adapts to a situation, when also the input vector follows a dimension reducing model.     

Diagnostic Measures for Generalized Linear Models with Missing Covariates

Abstract.  In this paper, we carry out an in-depth investigation of diagnostic measures for assessing the influence of observations and model misspecification in the presence of missing covariate data for generalized linear models. Our diagnostic measures include case-deletion measures and conditional residuals. We use the conditional residuals to construct goodness-of-fit statistics for testing possible misspecifications in model assumptions, including the sampling distribution. We develop specific strategies for incorporating missing data into goodness-of-fit statistics in order to increase the power of detecting model misspecification. A resampling method is proposed to approximate the p -value of the goodness-of-fit statistics. Simulation studies are conducted to evaluate our methods and a real data set is analysed to illustrate the use of our various diagnostic measures.     

Generalized Linear Factor Models: A New Local EM Estimation Algorithm

In this article, a general approach to latent variable models based on an underlying generalized linear model (GLM) with factor analysis observation process is introduced. We call these models Generalized Linear Factor Models (GLFM). The observations are produced from a general model framework that involves observed and latent variables that are assumed to be distributed in the exponential family. More specifically, we concentrate on situations where the observed variables are both discretely measured (e.g., binomial, Poisson) and continuously distributed (e.g., gamma). The common latent factors are assumed to be independent with a standard multivariate normal distribution. Practical details of training such models with a new local expectation-maximization (EM) algorithm, which can be considered as a generalized EM-type algorithm, are also discussed. In conjunction with an approximated version of the Fisher score algorithm (FSA), we show how to calculate maximum likelihood estimates of the model parameters, and to yield inferences about the unobservable path of the common factors. The methodology is illustrated by an extensive Monte Carlo simulation study and the results show promising performance.