局部平稳部分线性面板数据模型的统计推断

部分线性模型是一类非常重要的半参数回归模型,由于它既含有参数部分又含有非参数部分,与常规的线性模型相比具有更强的适应性和解释能力。文章研究带有局部平稳协变量的固定效应部分线性面板数据模型的统计推断。首先提出一个两阶段估计方法得到模型中未知参数和非参数函数的估计,并证明估计量的渐近性质,然后运用不变原理构造出非参数函数的一致置信带,最后通过数值模拟研究和实例分析验证了该方法的有效性。     

Treatment design selection effects on parameter estimation in dynamic logistic models for longitudinal binary data

In a longitudinal set-up, to examine the effects of certain fixed covariates on the repeated binary responses, there exists an approach to model the binary probabilities through a dynamic logistic relationship. In some practical situations such as in longitudinal clinical studies, it may happen that some of the covariates such as treatments are selected randomly following an adaptive design, whereas the rest of the covariates may be fixed by nature. The purpose of this study is to examine the effects of the design weights selection on the parameter estimation including the treatment effects, after taking the longitudinal correlations of the repeated binary responses into account.     

Inference in Semi‐Parametric Dynamic Models for Repeated Count Data

This paper deals with a longitudinal semi‐parametric regression model in a generalised linear model setup for repeated count data collected from a large number of independent individuals. To accommodate the longitudinal correlations, we consider a dynamic model for repeated counts which has decaying auto‐correlations as the time lag increases between the repeated responses. The semi‐parametric regression function involved in the model contains a specified regression function in some suitable time‐dependent covariates and a non‐parametric function in some other time‐dependent covariates. As far as the inference is concerned, because the non‐parametric function is of secondary interest, we estimate this function consistently using the independence assumption‐based well‐known quasi‐likelihood approach. Next, the proposed longitudinal correlation structure and the estimate of the non‐parametric function are used to develop a semi‐parametric generalised quasi‐likelihood approach for consistent and efficient estimation of the regression effects in the parametric regression function. The finite sample performance of the proposed estimation approach is examined through an intensive simulation study based on both large and small samples. Both balanced and unbalanced cluster sizes are incorporated in the simulation study. The asymptotic performances of the estimators are given. The estimation methodology is illustrated by reanalysing the well‐known health care utilisation data consisting of counts of yearly visits to a physician by 180 individuals for four years and several important primary and secondary covariates.     

Multiple imputation of a randomly censored covariate improves logistic regression analysis

Randomly censored covariates arise frequently in epidemiologic studies. The most commonly used methods, including complete case and single imputation or substitution, suffer from inefficiency and bias. They make strong parametric assumptions or they consider limit of detection censoring only. We employ multiple imputation, in conjunction with semi-parametric modeling of the censored covariate, to overcome these shortcomings and to facilitate robust estimation. We develop a multiple imputation approach for randomly censored covariates within the framework of a logistic regression model. We use the non-parametric estimate of the covariate distribution or the semi-parametric Cox model estimate in the presence of additional covariates in the model. We evaluate this procedure in simulations, and compare its operating characteristics to those from the complete case analysis and a survival regression approach. We apply the procedures to an Alzheimer's study of the association between amyloid positivity and maternal age of onset of dementia. Multiple imputation achieves lower standard errors and higher power than the complete case approach under heavy and moderate censoring and is comparable under light censoring. The survival regression approach achieves the highest power among all procedures, but does not produce interpretable estimates of association. Multiple imputation offers a favorable alternative to complete case analysis and ad hoc substitution methods in the presence of randomly censored covariates within the framework of logistic regression.     

Generation of pseudo random numbers and estimation under Cox models with time-dependent covariates

ABSTRACT

We study the method for generating pseudo random numbers under various special cases of the Cox model with time-dependent covariates when the baseline hazard function may not be constant and the random variable may equal infinity with a positive probability. During our simulation studies in computing the partial likelihood estimates, in between 3% and 20% of the time with a moderate sample size, it happens that the partial likelihood estimate of the regression coefficient is ∞ for the data from the Cox model. We propose a semi-parametric estimator as a modification for such a case. We present simulation results on the asymptotic properties of the semi-parametric estimator.     

A fully nonparametric approach in survival models with explanatory variables

The most widely used model for multidimensional survival analysis is the Cox model. This model is semi-parametric, since its hazard function is the product of an unspecified baseline hazard, and a parametric functional form relating the hazard and the covariates. We consider a more flexible and fully nonparametric proportional hazards model, where the functional form of the covariates effect is left unspecified. In this model, estimation is based on the maximum likelihood method. Results obtained from a Monte-Carlo experiment and from real data are presented. Finally, the advantages and the limitations of the approacha are discussed.     

Parametric versus semi-parametric models for the analysis of correlated survival data: A case study in veterinary epidemiology

Correlated survival data arise frequently in biomedical and epidemiologic research, because each patient may experience multiple events or because there exists clustering of patients or subjects, such that failure times within the cluster are correlated. In this paper, we investigate the appropriateness of the semi-parametric Cox regression and of the generalized estimating equations as models for clustered failure time data that arise from an epidemiologic study in veterinary medicine. The semi-parametric approach is compared with a proposed fully parametric frailty model. The frailty component is assumed to follow a gamma distribution. Estimates of the fixed covariates effects were obtained by maximizing the likelihood function, while an estimate of the variance component ( frailty parameter) was obtained from a profile likelihood construction.     

Robust signed-rank estimation and variable selection for semi-parametric additive partial linear models

A fully nonparametric model may not perform well or when the researcher wants to use a parametric model but the functional form with respect to a subset of the regressors or the density of the errors is not known. This becomes even more challenging when the data contain gross outliers or unusual observations. However, in practice the true covariates are not known in advance, nor is the smoothness of the functional form. A robust model selection approach through which we can choose the relevant covariates components and estimate the smoothing function may represent an appealing tool to the solution. A weighted signed-rank estimation and variable selection under the adaptive lasso for semi-parametric partial additive models is considered in this paper. B-spline is used to estimate the unknown additive nonparametric function. It is shown that despite using B-spline to estimate the unknown additive nonparametric function, the proposed estimator has an oracle property. The robustness of the weighted signed-rank approach for data with heavy-tail, contaminated errors, and data containing high-leverage points are validated via finite sample simulations. A practical application to an economic study is provided using an updated Canadian household gasoline consumption data.     

A consistent test for heteroscedasticity in semi-parametric regression with nonparametric variance function based on the kernel method

It is important to detect the variance heterogeneity in regression models. Heteroscedasticity tests have been well studied in parametric and nonparametric regression models. This paper presents a consistent test for heteroscedasticity for nonlinear semi-parametric regression models with nonparametric variance function based on the kernel method. The properties of the test are investigated through Monte Carlo simulations. The test methods are illustrated with a real example.     

Semiparametric multinomial logit models for analysing consumer choice behaviour

The multinomial logit model (MNL) is one of the most frequently used statistical models in marketing applications. It allows one to relate an unordered categorical response variable, for example representing the choice of a brand, to a vector of covariates such as the price of the brand or variables characterising the consumer. In its classical form, all covariates enter in strictly parametric, linear form into the utility function of the MNL model. In this paper, we introduce semiparametric extensions, where smooth effects of continuous covariates are modelled by penalised splines. A mixed model representation of these penalised splines is employed to obtain estimates of the corresponding smoothing parameters, leading to a fully automated estimation procedure. To validate semiparametric models against parametric models, we utilise different scoring rules as well as predicted market share and compare parametric and semiparametric approaches for a number of brand choice data sets.     

Estimation and specification testing in female labor participation models: parametric and semiparametric methods

Female labor participation models have been usually studied through probit and logit specifications. Little attention has been paid to verify the assumptions that are used in these sort of models, basically distributional assumptions and homoskedasticity. In this paper we apply semiparametirc methods in order to test the previous hypothesis. We also estimate a Spanish female labor participation model using both parametric and semiparametirc approaches. The parametirc model includes fixed and random coefficients probit specification. The estimation procedures are parametric maximum likelihood for both probit and logit models, and semiparametric quasi maximum likelihood following Klein and Spady (1993). The results depend cricially in the assumed model.     

A comparison of parametric and semi-parametric survival models with artificial neural networks

Survival models are used to examine data in the event of an occurrence. These are discussed in various types including parametric, non-parametric and semi-parametric models. Parametric models require a clear distribution of survival time, and semi-parametric models assume proportional hazards. Among these models, the non-parametric model of artificial neural network has the fewest assumptions and can be often replaced by other models. Given the importance of distribution Weibull survival models in this study of simulation shape parameter of the Weibull distribution have been assumed as 1, 2 and 3, and also the average rate at levels of 0%–75% have been censored. The values predicted by the neural network forecasting model with parametric survival and Cox regression models were compared. This comparison considering levels of complexity due to the hazard model using the ROC curve and the corresponding tests have been carried out.     

Bayesian Methods for Missing Covariates in Cure Rate Models

We propose methods for Bayesian inference for missing covariate data with a novel class of semi-parametric survival models with a cure fraction. We allow the missing covariates to be either categorical or continuous and specify a parametric distribution for the covariates that is written as a sequence of one dimensional conditional distributions. We assume that the missing covariates are missing at random (MAR) throughout. We propose an informative class of joint prior distributions for the regression coefficients and the parameters arising from the covariate distributions. The proposed class of priors are shown to be useful in recovering information on the missing covariates especially in situations where the missing data fraction is large. Properties of the proposed prior and resulting posterior distributions are examined. Also, model checking techniques are proposed for sensitivity analyses and for checking the goodness of fit of a particular model. Specifically, we extend the Conditional Predictive Ordinate (CPO) statistic to assess goodness of fit in the presence of missing covariate data. Computational techniques using the Gibbs sampler are implemented. A real data set involving a melanoma cancer clinical trial is examined to demonstrate the methodology.     

Modelling multi-stage processes through multivariate distributions

A new model combining parametric and semi-parametric approaches and following the lines of a semi-Markov model is developed for multi-stage processes. A Bivariate sojourn time distribution derived from the bivariate exponential distribution of Marshall & Olkin (1967) is adopted. The results compare favourably with the usual semi-parametric approaches that have been in use. Our approach also has several advantages over the models in use including its amenability to statistical inference. For example, the tests for symmetry and also for independence of the marginals of the sojourn time distributions, which were not available earlier, can now be conveniently derived and are enhanced in elegant forms. A unified Goodness-of-Fit test procedure for our proposed model is also presented. An application to the human resource planning involving real-life data from University of Nigeria is given.     

Estimating Optimal Dynamic Regimes: Correcting Bias under the Null

Abstract.  A dynamic regime provides a sequence of treatments that are tailored to patient-specific characteristics and outcomes. In 2004, James Robins proposed g –estimation using structural nested mean models (SNMMs) for making inference about the optimal dynamic regime in a multi-interval trial. The method provides clear advantages over traditional parametric approaches. Robins' g –estimation method always yields consistent estimators, but these can be asymptotically biased under a given SNMM for certain longitudinal distributions of the treatments and covariates, termed exceptional laws. In fact, under the null hypothesis of no treatment effect, every distribution constitutes an exceptional law under SNMMs which allow for interaction of current treatment with past treatments or covariates. This paper provides an explanation of exceptional laws and describes a new approach to g –estimation which we call Zeroing Instead of Plugging In (ZIPI). ZIPI provides nearly identical estimators to recursive g -estimators at non-exceptional laws while providing substantial reduction in the bias at an exceptional law when decision rule parameters are not shared across intervals.     

A new algorithm to estimate monotone nonparametric link functions and a comparison with parametric approach

The generalized linear model (GLM) is a class of regression models where the means of the response variables and the linear predictors are joined through a link function. Standard GLM assumes the link function is fixed, and one can form more flexible GLM by either estimating the flexible link function from a parametric family of link functions or estimating it nonparametically. In this paper, we propose a new algorithm that uses P-spline for nonparametrically estimating the link function which is guaranteed to be monotone. It is equivalent to fit the generalized single index model with monotonicity constraint. We also conduct extensive simulation studies to compare our nonparametric approach for estimating link function with various parametric approaches, including traditional logit, probit and robit link functions, and two recently developed link functions, the generalized extreme value link and the symmetric power logit link. The simulation study shows that the link function estimated nonparametrically by our proposed algorithm performs well under a wide range of different true link functions and outperforms parametric approaches when they are misspecified. A real data example is used to illustrate the results.     

Semi-parametric modelling and likelihood estimation with estimating equations

This paper proposes a semi-parametric modelling and estimating method for analysing censored survival data. The proposed method uses the empirical likelihood function to describe the information in data, and formulates estimating equations to incorporate knowledge of the underlying distribution and regression structure. The method is more flexible than the traditional methods such as the parametric maximum likelihood estimation (MLE), Cox's (1972) proportional hazards model, accelerated life test model, quasi-likelihood (Wedderburn, 1974) and generalized estimating equations (Liang & Zeger, 1986). This paper shows the existence and uniqueness of the proposed semi-parametric maximum likelihood estimates (SMLE) with estimating equations. The method is validated with known cases studied in the literature. Several finite sample simulation and large sample efficiency studies indicate that when the sample size is larger than 100 the SMLE is compatible with the parametric MLE; and in all case studies, the SMLE is about 15% better than the parametric MLE with a mis-specified underlying distribution.     

A semi-parametric estimation of a survival function from incomplete and doubly censored data

The purpose of this paper is to present a semi-parametric estimation of a survival function when analyzing incomplete and doubly censored data. Under the assumption that the chance of censoring is not related to the individual's survivorship, we propose a consistent estimation of survival. The derived estimator treats the uncensored observations nonparametrically and uses parametric models for both right and left censored data. Some asymptotic properties and simulation studies are also presented in order to analyze the behavior of the proposed estimator.     

Quasi-Likelihood Regression with Multiple Indices and Smooth Link and Variance Functions

Abstract.  A flexible semi-parametric regression model is proposed for modelling the relationship between a response and multivariate predictor variables. The proposed multiple-index model includes smooth unknown link and variance functions that are estimated non-parametrically. Data-adaptive methods for automatic smoothing parameter selection and for the choice of the number of indices M are considered. This model adapts to complex data structures and provides efficient adaptive estimation through the variance function component in the sense that the asymptotic distribution is the same as if the non-parametric components are known. We develop iterative estimation schemes, which include a constrained projection method for the case where the regression parameter vectors are mutually orthogonal. The proposed methods are illustrated with the analysis of data from a growth bioassay and a reproduction experiment with medflies. Asymptotic properties of the estimated model components are also obtained.     

Interval-valued data regression using partial linear model

Semi-parametric modelling of interval-valued data is of great practical importance, as exampled by applications in economic and financial data analysis. We propose a flexible semi-parametric modelling of interval-valued data by integrating the partial linear regression model based on the Center & Range method, and investigate its estimation procedure. Furthermore, we introduce a test statistic that allows one to decide between a parametric linear model and a semi-parametric model, and approximate its null asymptotic distribution based on wild Bootstrap method to obtain the critical values. Extensive simulation studies are carried out to evaluate the performance of the proposed methodology and the new test. Moreover, several empirical data sets are analysed to document its practical applications.