Model selection of generalized partially linear models with missing covariates

In this paper, a generalized partially linear model (GPLM) with missing covariates is studied and a Monte Carlo EM (MCEM) algorithm with penalized-spline (P-spline) technique is developed to estimate the regression coefficients and nonparametric function, respectively. As classical model selection procedures such as Akaike's information criterion become invalid for our considered models with incomplete data, some new model selection criterions for GPLMs with missing covariates are proposed under two different missingness mechanism, say, missing at random (MAR) and missing not at random (MNAR). The most attractive point of our method is that it is rather general and can be extended to various situations with missing observations based on EM algorithm, especially when no missing data involved, our new model selection criterions are reduced to classical AIC. Therefore, we can not only compare models with missing observations under MAR/MNAR settings, but also can compare missing data models with complete-data models simultaneously. Theoretical properties of the proposed estimator, including consistency of the model selection criterions are investigated. A simulation study and a real example are used to illustrate the proposed methodology.     

Multivariate Birnbaum–Saunders regression model

In this paper, we propose a multivariate log-linear Birnbaum–Saunders regression model. We discuss maximum-likelihood estimation of the model parameters and provide closed-form expressions for the score function and for Fisher's information matrix. Hypothesis testing is performed using approximations obtained from the asymptotic normality of the maximum-likelihood estimator. Some influence methods, such as the local influence and generalized leverage are discussed and the normal curvatures for studying local influence are derived under some perturbation schemes. Further, a test for the homogeneity of the shape parameter of the multivariate regression model is investigated. A real data set is presented for illustrative purposes.     

Diagnostics for generalized Poisson regression models with errors in variables

In this paper, we develop diagnostic methods for generalized Poisson regression (GPR) models with errors in variables based on the corrected likelihood. The one-step approximations of the estimates in the case-deletion model are given and case-deletion and local influence measures are presented. Meanwhile, based on a corrected score function, the testing statistics for the significance of dispersion parameters in GPR models with measurement errors are investigated. Finally, illustration of our methodology is given through numerical examples.     

Nonnull asymptotic distributions of the LR,Wald, score and gradient statistics in generalized linear models with dispersion covariates

The class of generalized linear models with dispersion covariates, which allows us to jointly model the mean and dispersion parameters, is a natural extension to the classical generalized linear models. In this paper, we derive the asymptotic expansions under a sequence of Pitman alternatives (up to order n ^?1/2) for the nonnull distribution functions of the likelihood ratio, Wald, Rao score and gradient statistics in this class of models. The asymptotic distributions of these statistics are obtained for testing a subset of regression parameters and for testing a subset of dispersion parameters. Based on these nonnull asymptotic expansions, the power of all four tests, which are equivalent to first order, are compared. Furthermore, we consider Monte Carlo simulations in order to compare the finite-sample performance of these tests in this class of models. We present two empirical applications to two real data sets for illustrative purposes.     

Robust estimation of covariance parameters in partial linear model for longitudinal data

For longitudinal data, the within-subject dependence structure and covariance parameters may be of practical and theoretical interests. The estimation of covariance parameters has received much attention and been studied mainly in the framework of generalized estimating equations (GEEs). The GEEs method, however, is sensitive to outliers. In this paper, an alternative set of robust generalized estimating equations for both the mean and covariance parameters are proposed in the partial linear model for longitudinal data. The asymptotic properties of the proposed estimators of regression parameters, non-parametric function and covariance parameters are obtained. Simulation studies are conducted to evaluate the performance of the proposed estimators under different contaminations. The proposed method is illustrated with a real data analysis.     

Score test for testing zero-inflated Poisson regression against zero-inflated generalized Poisson alternatives

In several cases, count data often have excessive number of zero outcomes. This zero-inflated phenomenon is a specific cause of overdispersion, and zero-inflated Poisson regression model (ZIP) has been proposed for accommodating zero-inflated data. However, if the data continue to suggest additional overdispersion, zero-inflated negative binomial (ZINB) and zero-inflated generalized Poisson (ZIGP) regression models have been considered as alternatives. This study proposes the score test for testing ZIP regression model against ZIGP alternatives and proves that it is equal to the score test for testing ZIP regression model against ZINB alternatives. The advantage of using the score test over other alternative tests such as likelihood ratio and Wald is that the score test can be used to determine whether a more complex model is appropriate without fitting the more complex model. Applications of the proposed score test on several datasets are also illustrated.     

TAR模型加权秩估计及其性质讨论

秩估计是上个世纪60年代逐渐兴起的一种非参数方法,由于它具有稳健性等特征,从而得到较为广泛的应用。本文主要讨论了TAR模型随机加权秩估计及其性质问题,证明了基于一般计分函数的线性秩统计量关于回归参数的渐近一致线性性。本文讨论的建立在计分规则基础上的秩估计方法,虽然以TAR模型为对象,但其基本原理同样可以应用到其他非线性模型的参数估计中。     

Estimated estimating equations: semiparametric inference for clustered and longitudinal data

Summary.  We introduce a flexible marginal modelling approach for statistical inference for clustered and longitudinal data under minimal assumptions. This estimated estimating equations approach is semiparametric and the proposed models are fitted by quasi-likelihood regression, where the unknown marginal means are a function of the fixed effects linear predictor with unknown smooth link, and variance–covariance is an unknown smooth function of the marginal means. We propose to estimate the nonparametric link and variance–covariance functions via smoothing methods, whereas the regression parameters are obtained via the estimated estimating equations. These are score equations that contain nonparametric function estimates. The proposed estimated estimating equations approach is motivated by its flexibility and easy implementation. Moreover, if data follow a generalized linear mixed model, with either a specified or an unspecified distribution of random effects and link function, the model proposed emerges as the corresponding marginal (population-average) version and can be used to obtain inference for the fixed effects in the underlying generalized linear mixed model, without the need to specify any other components of this generalized linear mixed model. Among marginal models, the estimated estimating equations approach provides a flexible alternative to modelling with generalized estimating equations. Applications of estimated estimating equations include diagnostics and link selection. The asymptotic distribution of the proposed estimators for the model parameters is derived, enabling statistical inference. Practical illustrations include Poisson modelling of repeated epileptic seizure counts and simulations for clustered binomial responses.     

Period analysis of variable stars by robust smoothing

Summary.  The objective is to estimate the period and the light curve (or periodic function) of a variable star. Previously, several methods have been proposed to estimate the period of a variable star, but they are inaccurate especially when a data set contains outliers. We use a smoothing spline regression to estimate the light curve given a period and then find the period which minimizes the generalized cross-validation (GCV). The GCV method works well, matching an intensive visual examination of a few hundred stars, but the GCV score is still sensitive to outliers. Handling outliers in an automatic way is important when this method is applied in a 'data mining' context to a vary large star survey. Therefore, we suggest a robust method which minimizes a robust cross-validation criterion induced by a robust smoothing spline regression. Once the period has been determined, a nonparametric method is used to estimate the light curve. A real example and a simulation study suggest that the robust cross-validation and GCV methods are superior to existing methods.     

Beta Regression for Modelling Rates and Proportions

This paper proposes a regression model where the response is beta distributed using a parameterization of the beta law that is indexed by mean and dispersion parameters. The proposed model is useful for situations where the variable of interest is continuous and restricted to the interval (0, 1) and is related to other variables through a regression structure. The regression parameters of the beta regression model are interpretable in terms of the mean of the response and, when the logit link is used, of an odds ratio, unlike the parameters of a linear regression that employs a transformed response. Estimation is performed by maximum likelihood. We provide closed-form expressions for the score function, for Fisher's information matrix and its inverse. Hypothesis testing is performed using approximations obtained from the asymptotic normality of the maximum likelihood estimator. Some diagnostic measures are introduced. Finally, practical applications that employ real data are presented and discussed.     

Linear trimmed means for the linear regression with AR(1) errors model

For the linear regression with AR(1) errors model, the robust generalized and feasible generalized estimators of Lai et al. (2003) of regression parameters are shown to have the desired property of a robust Gauss Markov theorem. This is done by showing that these two estimators are the best among classes of linear trimmed means. Monte Carlo and data analysis for this technique have been performed.     

Estimation and testing in generalized partial linear models—A comparative study

A particular semiparametric model of interest is the generalized partial linear model (GPLM) which extends the generalized linear model (GLM) by a nonparametric component.The paper reviews different estimation procedures based on kernel methods as well as test procedures on the correct specification of this model (vs. a parametric generalized linear model). Simulations and an application to a data set on East–West German migration illustrate similarities and dissimilarities of the estimators and test statistics.     

Deletion diagnostics for generalized linear models using the adjusted Poisson likelihood function

In this article, we propose two novel diagnostic measures for the deletion of influential observations for regression parameters in the setting of generalized linear models. The proposed diagnostic methods are capable for detecting the influential observations under model misspecification, as long as the true underlying distributions have finite second moments.More specifically, it is demonstrated that the Poisson likelihood function can be properly adjusted to become asymptotically valid for practically all underlying discrete distributions. The adjusted Poisson regression model that achieves the robustness property is presented. Simulation studies and an illustration are performed to demonstrate the efficacy of the two novel diagnostic procedures.     

A Simple Approach to Inference in Random Coefficient Models

Random coefficient regression models have been used to analyze cross-sectional and longitudinal data in economics and growth-curve data from biological and agricultural experiments. In the literature several estimators, including the ordinary least squares and the estimated generalized least squares (EGLS), have been considered for estimating the parameters of the mean model. Based on the asymptotic properties of the EGLS estimators, test statistics have been proposed for testing linear hypotheses involving the parameters of the mean model. An alternative estimator, the simple mean of the individual regression coefficients, provides estimation and hypothesis-testing procedures that are simple to compute and teach. The large sample properties of this simple estimator are shown to be similar to that of the EGLS estimator. The performance of the proposed estimator is compared with that of the existing estimators by Monte Carlo simulation.     

The Performance of a Robust Multistage Estimator in Nonlinear Regression with Heteroscedastic Errors

In this article, a robust multistage parameter estimator is proposed for nonlinear regression with heteroscedastic variance, where the residual variances are considered as a general parametric function of predictors. The motivation is based on considering the chi-square distribution for the calculated sample variance of the data. It is shown that outliers that are influential in nonlinear regression parameter estimates are not necessarily influential in calculating the sample variance. This matter persuades us, not only to robustify the estimate of the parameters of the models for both the regression function and the variance, but also to replace the sample variance of the data by a robust scale estimate.     

Some asymptotic results for hypothesis testing in multivariate failure time data

This paper presents a class of generalized Wald, generalized score and generalized likelihood ratio statistics for hypothesis testing and model selection for multivariate failure time data. These statistics are based on a marginal hazard model with a common baseline hazard function. The large sample distributions of these statistics are examined. It is shown that the proposed test statistics follow asymptotically a weighted sum of independent χ₁² distributions.     

Robust estimators and robust tests for the slightly contaminated stochastic logistic population models

This article introduces the robust indirect technique for the slightly contaminated stochastic logistic population models. Based on discrete sampled data with a fixed unit of time between two consecutive observations, we not only construct the robust indirect inference generalized method of moments (GMM) estimator for the model parameters, but also propose a likelihood-ratio-type indirect statistic and a robust indirect GMM saddle-point statistic for testing the parameters of interest. In addition, we develop the robust exponential tilting estimator and the robust exponential tilting test to improve their small sample performances. Finally, their finite-sample properties are studied through Monte Carlo experiments.     

Testing inference in variable dispersion beta regressions

The class of beta regression models proposed by Ferrari and Cribari-Neto [Beta regression for modelling rates and proportions, Journal of Applied Statistics 31 (2004), pp. 799–815] is useful for modelling data that assume values in the standard unit interval (0, 1). The dependent variable relates to a linear predictor that includes regressors and unknown parameters through a link function. The model is also indexed by a precision parameter, which is typically taken to be constant for all observations. Some authors have used, however, variable dispersion beta regression models, i.e., models that include a regression submodel for the precision parameter. In this paper, we show how to perform testing inference on the parameters that index the mean submodel without having to model the data precision. This strategy is useful as it is typically harder to model dispersion effects than mean effects. The proposed inference procedure is accurate even under variable dispersion. We present the results of extensive Monte Carlo simulations where our testing strategy is contrasted to that in which the practitioner models the underlying dispersion and then performs testing inference. An empirical application that uses real (not simulated) data is also presented and discussed.     

Quasi Bayesian likelihood

If the form of the distribution of data is unknown, the Bayesian method fails in the parametric inference because there is no posterior distribution of the parameter. In this paper, a theoretical framework of Bayesian likelihood is introduced via the Hilbert space method, which is free of the distributions of data and the parameter. The posterior distribution and posterior score function based on given inner products are defined and, consequently, the quasi posterior distribution and quasi posterior score function are derived, respectively, as the projections of the posterior distribution and posterior score function onto the space spanned by given estimating functions. In the space spanned by data, particularly, an explicit representation for the quasi posterior score function is obtained, which can be derived as a projection of the true posterior score function onto this space. The methods of constructing conservative quasi posterior score and quasi posterior log-likelihood are proposed. Some examples are given to illustrate the theoretical results. As an application, the quasi posterior distribution functions are used to select variables for generalized linear models. It is proved that, for linear models, the variable selections via quasi posterior distribution functions are equivalent to the variable selections via the penalized residual sum of squares or regression sum of squares.     

Ordered Regressions

There are often situations where two or more regression functions are ordered over a range of covariate values. In this paper, we develop efficient constrained estimation and testing procedures for such models. Specifically, necessary and sufficient conditions for ordering generalized linear regressions are given and shown to unify previous results obtained for simple linear regression, for polynomial regression and in the analysis of covariance models. We show that estimating the parameters of ordered linear regressions requires either quadratic programming or semi‐infinite programming, depending on the shape of the covariate space. A distance‐type test for order is proposed. Simulations demonstrate that the proposed methodology improves the mean square error and power compared with the usual, unconstrained, estimation and testing procedures. Improvements are often substantial. The methodology is extended to order generalized linear models where convex semi‐infinite programming plays a role. The methodology is motivated by, and applied to, a hearing loss study.