A note on fiducial generalized pivots for in one-way heteroscedastic ANOVA with random effects

The paper deals with generalized confidence intervals for the between-group variance in one-way heteroscedastic (unbalanced) ANOVA with random effects. The approach used mimics the standard one applied in mixed linear models with two variance components, where interval estimators are based on a minimal sufficient statistic derived after an initial reduction by the principle of invariance. A minimal sufficient statistic under heteroscedasticity is found to resemble its homoscedastic counterpart and further analogies between heteroscedastic and homoscedastic cases lead us to two classes of fiducial generalized pivots for the between-group variance. The procedures suggested formerly by Wimmer and Witkovský [Between group variance component interval estimation for the unbalanced heteroscedastic one-way random effects model, J. Stat. Comput. Simul. 73 (2003), pp. 333–346] and Li [Comparison of confidence intervals on between group variance in unbalanced heteroscedastic one-way random models, Comm. Statist. Simulation Comput. 36 (2007), pp. 381–390] are found to belong to these two classes. We comment briefly on some of their properties that were not mentioned in the original papers. In addition, properties of another particular generalized pivot are considered.     

Robust Testing Procedures in Heteroscedastic Linear Models

There exist many studies which treat the robust tests in homoscedastic linear models. However, the robust testing procedure in heteroscedastic linear models has not been examined. In this article, three classes of testing procedures for testing subhypothesis in heteroscedastic linear models are developed. These are Wald-type, score-type, and drop-in dispersion tests. The asymptotic distributions of these tests are obtained under the null hypothesis and contiguous alternatives. For a robustness criterion, the maximum asymptotic bias of the level of the test for distributions in a shrinking contamination neighborhood is used and the most-efficient robust test is derived. Finally, the performance of these tests in small sample is studied by Monte Carlo simulation.     

Heteroscedastic log-exponentiated Weibull regression model

We introduce a new class of heteroscedastic log-exponentiated Weibull (LEW) regression models. The class of regression models can be applied to censored data and be used more effectively in survival analysis. Maximum likelihood estimation of the model parameters with censored data as well as influence diagnostics for the new regression model is investigated. For different parameter settings, sample sizes and censoring percentages, various simulation studies are performed and compared to the performance of the heteroscedastic LEW regression model. The normal curvatures for studying local influence are derived under various perturbation schemes. An empirical application to a real data set is provided to illustrate the usefulness of the new class of heteroscedastic regression models.     

A new look at the difference between the GEE and the GLMM when modeling longitudinal count responses

Poisson log-linear regression is a popular model for count responses. We examine two popular extensions of this model – the generalized estimating equations (GEE) and the generalized linear mixed-effects model (GLMM) – to longitudinal data analysis and complement the existing literature on characterizing the relationship between the two dueling paradigms in this setting. Unlike linear regression, the GEE and the GLMM carry significant conceptual and practical implications when applied to modeling count data. Our findings shed additional light on the differences between the two classes of models when used for count data. Our considerations are demonstrated by both real study and simulated data.     

Efficient Estimation in Marginal Partially Linear Models for Longitudinal/Clustered Data Using Splines

Abstract.  We consider marginal semiparametric partially linear models for longitudinal/clustered data and propose an estimation procedure based on a spline approximation of the non-parametric part of the model and an extension of the parametric marginal generalized estimating equations (GEE). Our estimates of both parametric part and non-parametric part of the model have properties parallel to those of parametric GEE, that is, the estimates are efficient if the covariance structure is correctly specified and they are still consistent and asymptotically normal even if the covariance structure is misspecified. By showing that our estimate achieves the semiparametric information bound, we actually establish the efficiency of estimating the parametric part of the model in a stronger sense than what is typically considered for GEE. The semiparametric efficiency of our estimate is obtained by assuming only conditional moment restrictions instead of the strict multivariate Gaussian error assumption.     

Inference for linear and nonlinear stable error processes via estimating functions

This paper describes an estimating function approach for parameter estimation in linear and nonlinear times series models with infinite variance stable errors. Joint estimates of location and scale parameters are derived for classes of autoregressive (AR) models and random coefficient autoregressive (RCA) models with stable errors, as well as for AR models with stable autoregressive conditionally heteroscedastic (ARCH) errors. Fast, on-line, recursive parametric estimation for the location parameter based on estimating functions is discussed using simulation studies. A real financial time series is also discussed in some detail.     

Statistical inference in the partial linear models with the inverse gaussian kernel

Symmetric kernel smoothing is commonly used in estimating the nonparametric component in the partial linear regression models. In this article, we propose a new estimation method for the partial linear regression models using the inverse Gaussian kernel when the explanatory variable of the nonparametric component is non-negatively supported. As an asymmetric kernel function, the inverse Gaussian kernel is also supported on the non-negative half line. The asymptotic properties, including the asymptotic normality, uniform almost sure convergence, and the iterated logarithm laws, of the proposed estimators are thoroughly discussed for both homoscedastic and heteroscedastic cases. The simulation study is conducted to evaluate the finite sample performance of the proposed estimators.     

Risk Comparison of Inequality Constrained Estimators in the Heteroscedastic Linear Model

There exist many studies which treat the inequality and/or interval constraints on coefficients in the homoscedastic linear regression model. However, the sampling performance of the inequality constrained estimators in the heteroscedastic linear model has not been examined. This paper considers the inequality constrained estimators in the heteroscedastic linear regression model and derives their risks under a quadratic loss function. Furthermore, using the inequality constrained estimators, we introduce a pre-test estimator which might be employed after the test for homoscedasticity and derive its risk. In addition, the risk performance of these estimators is evaluated numerically.     

A flexible semiparametric regression model for bimodal,asymmetric and censored data

In this paper, we propose a new semiparametric heteroscedastic regression model allowing for positive and negative skewness and bimodal shapes using the B-spline basis for nonlinear effects. The proposed distribution is based on the generalized additive models for location, scale and shape framework in order to model any or all parameters of the distribution using parametric linear and/or nonparametric smooth functions of explanatory variables. We motivate the new model by means of Monte Carlo simulations, thus ignoring the skewness and bimodality of the random errors in semiparametric regression models, which may introduce biases on the parameter estimates and/or on the estimation of the associated variability measures. An iterative estimation process and some diagnostic methods are investigated. Applications to two real data sets are presented and the method is compared to the usual regression methods.     

Bayesian inference in a heteroscedastic replicated measurement error model using heavy-tailed distributions

We introduce a multivariate heteroscedastic measurement error model for replications under scale mixtures of normal distribution. The model can provide a robust analysis and can be viewed as a generalization of multiple linear regression from both model structure and distribution assumption. An efficient method based on Markov Chain Monte Carlo is developed for parameter estimation. The deviance information criterion and the conditional predictive ordinates are used as model selection criteria. Simulation studies show robust inference behaviours of the model against both misspecification of distributions and outliers. We work out an illustrative example with a real data set on measurements of plant root decomposition.     

Statistical Inference for the Multidimensional Mixed Rasch Model

Inference in generalized linear mixed models with multivariate random effects is often made cumbersome by the high-dimensional intractable integrals involved in the marginal likelihood. This article presents an inferential methodology based on the GEE approach. This method involves the approximations of the marginal likelihood and joint moments of the variables. It is also proposed an approximate Akaike and Bayesian information criterions based on the approximate marginal likelihood using the estimation of the parameters by the GEE approach. The different results are illustrated with a simulation study and with an analysis of real data from health-related quality of life.     

Goodness-of-fit test for Gaussian regression with block correlated errors

We propose a procedure to test that the expectation of a Gaussian vector is linear against a nonparametric alternative. We consider the case where the covariance matrix of the observations has a block diagonal structure. This framework encompasses regression models with autocorrelated errors, heteroscedastic regression models, mixed-effects models and growth curves. Our procedure does not depend on any prior information about the alternative. We prove that the test is asymptotically of the nominal level and consistent. We characterize the set of vectors on which the test is powerful and prove the classical √log log (n)/n convergence rate over directional alternatives. We propose a bootstrap version of the test as an alternative to the initial one and provide a simulation study in order to evaluate both procedures for small sample sizes when the purpose is to test goodness of fit in a Gaussian mixed-effects model. Finally, we illustrate the procedures using a real data set.     

Estimation of the Variance Function in Heteroscedastic Linear Regression Models

Heteroscedasticity generally exists when a linear regression model is applied to analyzing some real-world problems. Therefore, how to accurately estimate the variance functions of the error term in a heteroscedastic linear regression model is of great importance for obtaining efficient estimates of the regression parameters and making valid statistical inferences. A method for estimating the variance function of heteroscedastic linear regression models is proposed in this article based on the variance-reduced local linear smoothing technique. Some simulations and comparisons with other method are conducted to assess the performance of the proposed method. The results demonstrate that the proposed method can accurately estimate the variance functions and therefore produce more efficient estimates of the regression parameters.     

Regularized estimation for highly multivariate log Gaussian Cox processes

Statistical inference for highly multivariate point pattern data is challenging due to complex models with large numbers of parameters. In this paper, we develop numerically stable and efficient parameter estimation and model selection algorithms for a class of multivariate log Gaussian Cox processes. The methodology is applied to a highly multivariate point pattern data set from tropical rain forest ecology.     

Tuning Parameter Selector for the Penalized Likelihood Method in Multivariate Generalized Linear Models

Variable selection is fundamental to high-dimensional multivariate generalized linear models. The smoothly clipped absolute deviation (SCAD) method can solve the problem of variable selection and estimation. The choice of the tuning parameter in the SCAD method is critical, which controls the complexity of the selected model. This article proposes a criterion to select the tuning parameter for the SCAD method in multivariate generalized linear models, which is shown to be able to identify the true model consistently. Simulation studies are conducted to support theoretical findings, and two real data analysis are given to illustrate the proposed method.     

Quantile regression in heteroscedastic varying coefficient models

Varying coefficient models are flexible models to describe the dynamic structure in longitudinal data. Quantile regression, more than mean regression, gives partial information on the conditional distribution of the response given the covariates. In the literature, the focus has been so far mostly on homoscedastic quantile regression models, whereas there is an interest in looking into heteroscedastic modelling. This paper contributes to the area by modelling the heteroscedastic structure and estimating it from the data, together with estimating the quantile functions. The use of the proposed methods is illustrated on real-data applications. The finite-sample behaviour of the methods is investigated via a simulation study, which includes a comparison with an existing method.     

Estimation of parameters in incomplete data models defined by dynamical systems

Parametric incomplete data models defined by ordinary differential equations (ODEs) are widely used in biostatistics to describe biological processes accurately. Their parameters are estimated on approximate models, whose regression functions are evaluated by a numerical integration method. Accurate and efficient estimations of these parameters are critical issues. This paper proposes parameter estimation methods involving either a stochastic approximation EM algorithm (SAEM) in the maximum likelihood estimation, or a Gibbs sampler in the Bayesian approach. Both algorithms involve the simulation of non-observed data with conditional distributions using Hastings–Metropolis (H–M) algorithms. A modified H–M algorithm, including an original local linearization scheme to solve the ODEs, is proposed to reduce the computational time significantly. The convergence on the approximate model of all these algorithms is proved. The errors induced by the numerical solving method on the conditional distribution, the likelihood and the posterior distribution are bounded. The Bayesian and maximum likelihood estimation methods are illustrated on a simulated pharmacokinetic nonlinear mixed-effects model defined by an ODE. Simulation results illustrate the ability of these algorithms to provide accurate estimates.     

A Multiple Imputation Approach to Linear Regression with Clustered Censored Data

Weextend Wei and Tanner's (1991) multiple imputation approach insemi-parametric linear regression for univariate censored datato clustered censored data. The main idea is to iterate the followingtwo steps: 1) using the data augmentation to impute for censoredfailure times; 2) fitting a linear model with imputed completedata, which takes into consideration of clustering among failuretimes. In particular, we propose using the generalized estimatingequations (GEE) or a linear mixed-effects model to implementthe second step. Through simulation studies our proposal comparesfavorably to the independence approach (Lee et al., 1993), whichignores the within-cluster correlation in estimating the regressioncoefficient. Our proposal is easy to implement by using existingsoftwares.     

Modeling longitudinal binomial responses: implications from two dueling paradigms

The generalized estimating equations (GEEs) and generalized linear mixed-effects model (GLMM) are the two most popular paradigms to extend models for cross-sectional data to a longitudinal setting. Although the two approaches yield well-interpreted models for continuous outcomes, it is quite a different story when applied to binomial responses. We discuss major modeling differences between the GEE- and GLMM-derived models by presenting new results regarding the model-driven differences. Our results show that GLMM induces some artifacts in the marginal models at assessment times, making it inappropriate when applied to such responses from real study data. The different interpretations of parameters resulting from the conceptual difference between the two modeling approaches also carry quite significant implications and ramifications with respect to data and power analyses. Although a special case involving a scale difference in parameters between GEE and GLMM has been noted in the literature, its implications in real data analysis has not been thoroughly addressed. Further, this special case has a very limited covariate structure and does not apply to most real studies, especially multi-center clinical trials. The new results presented fill a substantial gap in the literature regarding the model-driven differences between the two dueling paradigms.