On partial linear additive isotonic regression

In this paper we discuss semiparametric additive isotonic regression models. We discuss the efficiency bound of the model and the least squares estimator under this model. We show that the ordinary least square estimator studied by Huang (2002) and Cheng (2009) for the semiparametric isotonic regression achieves the efficiency bound for the regular estimator when the true parameter belongs to the interior of the parameter space. We also show that the result by Cheng (2009) can be generalized to the case that the covariates are dependent on each other.     

Polynomial Spline Estimation for a Generalized Additive Coefficient Model

Abstract.  We study a semiparametric generalized additive coefficient model (GACM), in which linear predictors in the conventional generalized linear models are generalized to unknown functions depending on certain covariates, and approximate the non-parametric functions by using polynomial spline. The asymptotic expansion with optimal rates of convergence for the estimators of the non-parametric part is established. Semiparametric generalized likelihood ratio test is also proposed to check if a non-parametric coefficient can be simplified as a parametric one. A conditional bootstrap version is suggested to approximate the distribution of the test under the null hypothesis. Extensive Monte Carlo simulation studies are conducted to examine the finite sample performance of the proposed methods. We further apply the proposed model and methods to a data set from a human visceral Leishmaniasis study conducted in Brazil from 1994 to 1997. Numerical results outperform the traditional generalized linear model and the proposed GACM is preferable.     

On asymptotic properties of Bayesian partially linear models

In this paper, we present large sample properties of a partially linear model from the Bayesian perspective, in which responses are explained by the semiparametric regression model with the additive form of the linear component and the nonparametric component. For this purpose, we investigate asymptotic behaviors of posterior distributions in terms of consistency. Specifically, we deal with a specific Bayesian partially linear regression model with additive noises in which the nonparametric component is modeled using Gaussian process priors. Under the Bayesian partially linear model using Gaussian process priors, we focus on consistency of posterior distribution and consistency of the Bayes factor, and extend these results to generalized additive regression models and study their asymptotic properties. In addition we illustrate the asymptotic properties based on empirical analysis through simulation studies.     

Bayesian inference for generalized additive mixed models based on Markov random field priors

Most regression problems in practice require flexible semiparametric forms of the predictor for modelling the dependence of responses on covariates. Moreover, it is often necessary to add random effects accounting for overdispersion caused by unobserved heterogeneity or for correlation in longitudinal or spatial data. We present a unified approach for Bayesian inference via Markov chain Monte Carlo simulation in generalized additive and semiparametric mixed models. Different types of covariates, such as the usual covariates with fixed effects, metrical covariates with non-linear effects, unstructured random effects, trend and seasonal components in longitudinal data and spatial covariates, are all treated within the same general framework by assigning appropriate Markov random field priors with different forms and degrees of smoothness. We applied the approach in several case-studies and consulting cases, showing that the methods are also computationally feasible in problems with many covariates and large data sets. In this paper, we choose two typical applications.     

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.     

Additive Transformation Models for Clustered Doubly Censored Data

In this article, we propose estimating procedures for additive semiparametric transformation models with clustered doubly-censored data. A simulation study is conducted to investigate the performance of the proposed estimators. We apply the proposed methods to the dataset from the well-known Diabetic Retinopathy Study.     

SEMIPARAMETRIC ADDITIVE RISKS REGRESSION FOR TWO-STAGE DESIGN SURVIVAL STUDIES

In this article we study a semiparametric additive risks model (McKeague and Sasieni (1994)) for two-stage design survival data where accurate information is available only on second stage subjects, a subset of the first stage study. We derive two-stage estimators by combining data from both stages. Large sample inferences are developed. As a by-product, we also obtain asymptotic properties of the single stage estimators of McKeague and Sasieni (1994) when the semiparametric additive risks model is misspecified. The proposed two-stage estimators are shown to be asymptotically more efficient than the second stage estimators. They also demonstrate smaller bias and variance for finite samples. The developed methods are illustrated using small intestine cancer data from the SEER (Surveillance, Epidemiology, and End Results) Program.     

A new semiparametric Weibull cure rate model: fitting different behaviors within GAMLSS

ABSTRACT

We propose a new semiparametric Weibull cure rate model for fitting nonlinear effects of explanatory variables on the mean, scale and cure rate parameters. The regression model is based on the generalized additive models for location, scale and shape, for which any or all distribution parameters can be modeled as parametric linear and/or nonparametric smooth functions of explanatory variables. We present methods to select additive terms, model estimation and validation, where all computational codes are presented in a simple way such that any R user can fit the new model. Biases of the parameter estimates caused by models specified erroneously are investigated through Monte Carlo simulations. We illustrate the usefulness of the new model by means of two applications to real data. We provide computational codes to fit the new regression model in the R software.     

Efficient Estimation of the Partly Linear Additive Hazards Model with Current Status Data

This paper focuses on efficient estimation, optimal rates of convergence and effective algorithms in the partly linear additive hazards regression model with current status data. We use polynomial splines to estimate both cumulative baseline hazard function with monotonicity constraint and nonparametric regression functions with no such constraint. We propose a simultaneous sieve maximum likelihood estimation for regression parameters and nuisance parameters and show that the resultant estimator of regression parameter vector is asymptotically normal and achieves the semiparametric information bound. In addition, we show that rates of convergence for the estimators of nonparametric functions are optimal. We implement the proposed estimation through a backfitting algorithm on generalized linear models. We conduct simulation studies to examine the finite‐sample performance of the proposed estimation method and present an analysis of renal function recovery data for illustration.     

Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations

Summary.  Structured additive regression models are perhaps the most commonly used class of models in statistical applications. It includes, among others, (generalized) linear models, (generalized) additive models, smoothing spline models, state space models, semiparametric regression, spatial and spatiotemporal models, log-Gaussian Cox processes and geostatistical and geoadditive models. We consider approximate Bayesian inference in a popular subset of structured additive regression models, latent Gaussian models , where the latent field is Gaussian, controlled by a few hyperparameters and with non-Gaussian response variables. The posterior marginals are not available in closed form owing to the non-Gaussian response variables. For such models, Markov chain Monte Carlo methods can be implemented, but they are not without problems, in terms of both convergence and computational time. In some practical applications, the extent of these problems is such that Markov chain Monte Carlo sampling is simply not an appropriate tool for routine analysis. We show that, by using an integrated nested Laplace approximation and its simplified version, we can directly compute very accurate approximations to the posterior marginals. The main benefit of these approximations is computational: where Markov chain Monte Carlo algorithms need hours or days to run, our approximations provide more precise estimates in seconds or minutes. Another advantage with our approach is its generality, which makes it possible to perform Bayesian analysis in an automatic, streamlined way, and to compute model comparison criteria and various predictive measures so that models can be compared and the model under study can be challenged.     

Estimation of the mixtures of GLMs with covariate-dependent mixing proportions

ABSTRACT

In this article, we study the estimation for a class of semiparametric mixtures of generalized linear models where mixing proportions depend on a covariate non parametrically. We investigate a backfitting estimation procedure and show the asymptotic normality of the proposed estimators under mild conditions. We conduct simulation to show the good performance of our methodology and give a real data analysis as an illustration.     

A note on the asymptotic relative efficiency of estimators from the additive hazards model

In this note, the asymptotic variance formulas are explicitly derived and compared between the parametric and semiparametric estimators of a regression parameter and survival probability under the additive hazards model. To obtain explicit formulas, it is assumed that the covariate term including a regression coefficient follows a gamma distribution and the baseline hazard function is constant. The results show that the semiparametric estimator of the regression coefficient parameter is fully efficient relative to the parametric counterpart when the survival time and a covariate are independent, as in the proportional hazards model. Relative to a more realistic case of the parametric additive hazards model with a Weibull baseline, the loss of efficiency of the semiparametric estimator of survival probability is moderate.     

Generalized partially linear varying coefficient models with multiple smoothing variables

This paper is concerned with semiparametric efficient estimation of a generalized partially linear varying coefficient model. The model studied in this paper is very flexible, accommodating various nonlinear relations between the response variable and a set of predictor variables. It is a structured regression model and is particularly useful in dealing with a discrete response variable. We apply the smooth backfitting technique to estimate the nonparametric part of the model and employ the profiling approach to obtain a semiparametric efficient estimator of the parametric part.     

Pseudomartingale estimating equations for modulated renewal process models

Summary.  We adapt martingale estimating equations based on gap time information to a general intensity model for a single realization of a modulated renewal process. The consistency and asymptotic normality of the estimators is proved under ergodicity conditions. Previous work has considered either parametric likelihood analysis or semiparametric multiplicative models using partial likelihood. The framework is generally applicable to semiparametric and parametric models, including additive and multiplicative specifications, and periodic models. It facilitates a semiparametric extension of a popular parametric earthquake model. Simulations and empirical analyses of Taiwanese earthquake sequences illustrate the methodology's practical utility.     

A generalized semiparametric regression and its efficient estimation

We investigate a generalized semiparametric regression. Such a model can avoid the risk of wrongly choosing the base measure function. We propose a profile likelihood to efficiently estimate both parameter and nonparametric function. The main difference from the classical profile likelihood is that the profile likelihood proposed is a functional of the base measure function, instead of a function of a real variable. By making the most of the structure information of the semiparametric exponential family, we get an explicit expression of the estimator of the least favorable curve. It ensures that the new profile likelihood is computationally simple. Due to the use of the least favorable curve, the semiparametric efficiency is achieved successfully and the estimation bias is reduced significantly. Simulation studies can illustrate that our proposal is much better than the existing methodologies for most cases under study, and is robust to the different model conditions.     

Local and omnibus goodness-of-fit tests in classical measurement error models

We consider functional measurement error models, i.e. models where covariates are measured with error and yet no distributional assumptions are made about the mismeasured variable. We propose and study a score-type local test and an orthogonal series-based, omnibus goodness-of-fit test in this context, where no likelihood function is available or calculated-i.e. all the tests are proposed in the semiparametric model framework. We demonstrate that our tests have optimality properties and computational advantages that are similar to those of the classical score tests in the parametric model framework. The test procedures are applicable to several semiparametric extensions of measurement error models, including when the measurement error distribution is estimated non-parametrically as well as for generalized partially linear models. The performance of the local score-type and omnibus goodness-of-fit tests is demonstrated through simulation studies and analysis of a nutrition data set.     

Propriety of posteriors in structured additive regression models: Theory and empirical evidence

Structured additive regression comprises many semiparametric regression models such as generalized additive (mixed) models, geoadditive models, and hazard regression models within a unified framework. In a Bayesian formulation, non-parametric functions, spatial effects and further model components are specified in terms of multivariate Gaussian priors for high-dimensional vectors of regression coefficients. For several model terms, such as penalized splines or Markov random fields, these Gaussian prior distributions involve rank-deficient precision matrices, yielding partially improper priors. Moreover, hyperpriors for the variances (corresponding to inverse smoothing parameters) may also be specified as improper, e.g. corresponding to Jeffreys prior or a flat prior for the standard deviation. Hence, propriety of the joint posterior is a crucial issue for full Bayesian inference in particular if based on Markov chain Monte Carlo simulations. We establish theoretical results providing sufficient (and sometimes necessary) conditions for propriety and provide empirical evidence through several accompanying simulation studies.     

Estimating partially linear panel data models with one-way error components

We consider the problem of estimating a partially linear panel data model whenthe error follows an one-way error components structure. We propose a feasiblesemiparametric generalized least squares (GLS) type estimator for estimating the coefficient of the linear component and show that it is asymptotically more efficient than a semiparametric ordinary least squares (OLS) type estimator. We also discussed the case when the regressor of the parametric component is correlated with the error, and propose an instrumental variable GLS-type semiparametric estimator.     

Testing generalized linear and semiparametric models against smooth alternatives

We propose goodness-of-fit tests for testing generalized linear models and semiparametric regression models against smooth alternatives. The focus is on models having both continous and factorial covariates. As a smooth extension of a parametric or semiparametric model we use generalized varying-coefficient models as proposed by Hastie and Tibshirani. A likelihood ratio statistic is used for testing. Asymptotic expansions allow us to write the estimates as linear smoothers which in turn guarantees simple and fast bootstrapping of the test statistic. The test is shown to have √ n -power, but in contrast with parametric tests it is powerful against smooth alternatives in general.     

Bayesian spectral analysis models for quantile regression with Dirichlet process mixtures

This paper presents a Bayesian analysis of partially linear additive models for quantile regression. We develop a semiparametric Bayesian approach to quantile regression models using a spectral representation of the nonparametric regression functions and the Dirichlet process (DP) mixture for error distribution. We also consider Bayesian variable selection procedures for both parametric and nonparametric components in a partially linear additive model structure based on the Bayesian shrinkage priors via a stochastic search algorithm. Based on the proposed Bayesian semiparametric additive quantile regression model referred to as BSAQ, the Bayesian inference is considered for estimation and model selection. For the posterior computation, we design a simple and efficient Gibbs sampler based on a location-scale mixture of exponential and normal distributions for an asymmetric Laplace distribution, which facilitates the commonly used collapsed Gibbs sampling algorithms for the DP mixture models. Additionally, we discuss the asymptotic property of the sempiparametric quantile regression model in terms of consistency of posterior distribution. Simulation studies and real data application examples illustrate the proposed method and compare it with Bayesian quantile regression methods in the literature.