Quantile regression for robust inference on varying coefficient partially nonlinear models

In this paper, we propose a robust statistical inference approach for the varying coefficient partially nonlinear models based on quantile regression. A three-stage estimation procedure is developed to estimate the parameter and coefficient functions involved in the model. Under some mild regularity conditions, the asymptotic properties of the resulted estimators are established. Some simulation studies are conducted to evaluate the finite performance as well as the robustness of our proposed quantile regression method versus the well known profile least squares estimation procedure. Moreover, the Boston housing price data is given to further illustrate the application of the new method.     

Estimation for semiparametric varying coefficient models with different smoothing variables under random right censoring

In this paper we study a semiparametric varying coefficient model when the response is subject to random right censoring. The model gives an easy interpretation due to its direct connectivity to the classical linear model and is very flexible since nonparametric functions which accommodates various nonlinear interaction effects between covariates are admitted in the model. We propose estimators for this model using mean-preserving transformation and establish their asymptotic properties. The estimation procedure is based on the profiling and the smooth backfitting techniques. A simulation study is presented to show the reliability of the proposed estimators and an automatic bandwidth selector is given in a data-driven way.     

Convergence rates for smoothing spline estimators in varying coefficient models

We consider the estimation of a multiple regression model in which the coefficients change slowly in “time”, with “time” being an additional covariate. Under reasonable smoothness conditions, we prove the usual expected mean square error bounds for the smoothing spline estimators of the coefficient functions.     

Small area estimation of poverty indicators

The authors propose to estimate nonlinear small area population parameters by using the empirical Bayes (best) method, based on a nested error model. They focus on poverty indicators as particular nonlinear parameters of interest, but the proposed methodology is applicable to general nonlinear parameters. They use a parametric bootstrap method to estimate the mean squared error of the empirical best estimators. They also study small sample properties of these estimators by model‐based and design‐based simulation studies. Results show large reductions in mean squared error relative to direct area‐specific estimators and other estimators obtained by “simulated” censuses. The authors also apply the proposed method to estimate poverty incidences and poverty gaps in Spanish provinces by gender with mean squared errors estimated by the mentioned parametric bootstrap method. For the Spanish data, results show a significant reduction in coefficient of variation of the proposed empirical best estimators over direct estimators for practically all domains. The Canadian Journal of Statistics 38: 369–385; 2010 © 2010 Statistical Society of Canada     

Model Selection,Transformations and Variance Estimation in Nonlinear Regression

The results of analyzing experimental data using a parametric model may heavily depend on the chosen model for regression and variance functions, moreover also on a possibly underlying preliminary transformation of the variables. In this paper we propose and discuss a complex procedure which consists in a simultaneous selection of parametric regression and variance models from a relatively rich model class and of Box-Cox variable transformations by minimization of a cross-validation criterion. For this it is essential to introduce modifications of the standard cross-validation criterion adapted to each of the following objectives: 1. estimation of the unknown regression function, 2. prediction of future values of the response variable, 3. calibration or 4. estimation of some parameter with a certain meaning in the corresponding field of application. Our idea of a criterion oriented combination of procedures (which usually if applied, then in an independent or sequential way) is expected to lead to more accurate results. We show how the accuracy of the parameter estimators can be assessed by a “moment oriented bootstrap procedure", which is an essential modification of the “wild bootstrap” of Härdle and Mammen by use of more accurate variance estimates. This new procedure and its refinement by a bootstrap based pivot (“double bootstrap”) is also used for the construction of confidence, prediction and calibration intervals. Programs written in Splus which realize our strategy for nonlinear regression modelling and parameter estimation are described as well. The performance of the selected model is discussed, and the behaviour of the procedures is illustrated, e.g., by an application in radioimmunological assay.     

Estimating conditional tail expectation with actuarial applications in view

We develop statistical inferential tools for estimating and comparing conditional tail expectation (CTE) functions, which are of considerable interest in actuarial science. In particular, we construct estimators for the CTE functions, develop the necessary asymptotic theory for the estimators, and then use the theory for constructing confidence intervals and bands for the functions. Both parametric and non-parametric approaches are explored. Simulation studies illustrate the performance of estimators in various situations. Results are obtained under minimal assumptions, and the general Vervaat process plays a crucial role in achieving these goals.     

Reducing component estimation for varying coefficient models

The estimation problem for varying coefficient models has been studied by many authors. We consider the problem in the case that the unknown functions admit different degrees of smoothness. In this paper we propose a reducing component local polynomial method to estimate the unknown functions. It is shown that all of our estimators achieve the optimal convergence rates. The asymptotic distributions of our estimators are also derived. The established asymptotic results and the simulation results show that our estimators outperform the the existing two-step estimators when the coefficient functions admit different degrees of smoothness. We also develop methods to speed up the estimation of the model and the selection of the bandwidths.     

Robust Linear Calibration

We regard the simple linear calibration problem where only the response y of the regression line y = β₀ + β₁ t is observed with errors. The experimental conditions t are observed without error. For the errors of the observations y we assume that there may be some gross errors providing outlying observations. This situation can be modeled by a conditionally contaminated regression model. In this model the classical calibration estimator based on the least squares estimator has an unbounded asymptotic bias. Therefore we introduce calibration estimators based on robust one-step-M-estimators which have a bounded asymptotic bias. For this class of estimators we discuss two problems: The optimal estimators and their corresponding optimal designs. We derive the locally optimal solutions and show that the maximin efficient designs for non-robust estimation and robust estimation coincide.     

Partial functional linear regression

It is frequently the case that a response will be related to both a vector of finite length and a function-valued random variable as predictor variables. In this paper, we propose new estimators for the parameters of a partial functional linear model which explores the relationship between a scalar response variable and mixed-type predictors. Asymptotic properties of the proposed estimators are established and finite sample behavior is studied through a small simulation experiment.     

The Bernstein–von Mises theorem in semiparametric competing risks models

Semiparametric Bayesian models are nowadays a popular tool in event history analysis. An important area of research concerns the investigation of frequentist properties of posterior inference. In this paper, we propose novel semiparametric Bayesian models for the analysis of competing risks data and investigate the Bernstein–von Mises theorem for differentiable functionals of model parameters. The model is specified by expressing the cause-specific hazard as the product of the conditional probability of a failure type and the overall hazard rate. We take the conditional probability as a smooth function of time and leave the cumulative overall hazard unspecified. A prior distribution is defined on the joint parameter space, which includes a beta process prior for the cumulative overall hazard. We first develop the large-sample properties of maximum likelihood estimators by giving simple sufficient conditions for them to hold. Then, we show that, under the chosen priors, the posterior distribution for any differentiable functional of interest is asymptotically equivalent to the sampling distribution derived from maximum likelihood estimation. A simulation study is provided to illustrate the coverage properties of credible intervals on cumulative incidence functions.     

Estimation and inference for varying coefficient partially nonlinear models

In this paper, we extend the varying coefficient partially linear model to the varying coefficient partially nonlinear model in which the linear part of the varying coefficient partially linear model is replaced by a nonlinear function of the covariates. A profile nonlinear least squares estimation procedure for the parameter vector and the coefficient function vector of the varying coefficient partially nonlinear model is proposed and the asymptotic properties of the resulting estimators are established. We further propose a generalized likelihood ratio (GLR) test to check whether or not the varying coefficients in the model are constant. The asymptotic null distribution of the GLR statistic is derived and a residual-based bootstrap procedure is also suggested to derive the p-value of the GLR test. Some simulations are conducted to assess the performance of the proposed estimating and testing procedures and the results show that both the procedures perform well in finite samples. Furthermore, a real data example is given to demonstrate the application of the proposed model and its estimating and testing procedures.     

Bias-reduced extreme quantile estimators of Weibull tail-distributions

In this paper, we consider the problem of estimating an extreme quantile of a Weibull tail-distribution. The new extreme quantile estimator has a reduced bias compared to the more classical ones proposed in the literature. It is based on an exponential regression model that was introduced in Diebolt et al. [2007. Bias-reduced estimators of the Weibull-tail coefficient. Test, to appear]. The asymptotic normality of the extreme quantile estimator is established. We also introduce an adaptive selection procedure to determine the number of upper order statistics to be used. A simulation study as well as an application to a real data set is provided in order to prove the efficiency of the above-mentioned methods.     

Statistical inference for a semiparametric measurement error regression model with heteroscedastic errors

Efficient inference for regression models requires that the heteroscedasticity be taken into account. We consider statistical inference under heteroscedasticity in a semiparametric measurement error regression model, in which some covariates are measured with errors. This paper has multiple components. First, we propose a new method for testing the heteroscedasticity. The advantages of the proposed method over the existing ones are that it does not need any nonparametric estimation and does not involve any mismeasured variables. Second, we propose a new two-step estimator for the error variances if there is heteroscedasticity. Finally, we propose a weighted estimating equation-based estimator (WEEBE) for the regression coefficients and establish its asymptotic properties. Compared with existing estimators, the proposed WEEBE is asymptotically more efficient, avoids undersmoothing the regressor functions and requires less restrictions on the observed regressors. Simulation studies show that the proposed test procedure and estimators have nice finite sample performance. A real data set is used to illustrate the utility of our proposed methods.     

Partial covariate adjusted regression

Covariate adjusted regression (CAR) is a recently proposed adjustment method for regression analysis where both the response and predictors are not directly observed [?entürk, D., Müller, H.G., 2005. Covariate adjusted regression. Biometrika 92, 75–89]. The available data have been distorted by unknown functions of an observable confounding covariate. CAR provides consistent estimators for the coefficients of the regression between the variables of interest, adjusted for the confounder. We develop a broader class of partial covariate adjusted regression (PCAR) models to accommodate both distorted and undistorted (adjusted/unadjusted) predictors. The PCAR model allows for unadjusted predictors, such as age, gender and demographic variables, which are common in the analysis of biomedical and epidemiological data. The available estimation and inference procedures for CAR are shown to be invalid for the proposed PCAR model. We propose new estimators and develop new inference tools for the more general PCAR setting. In particular, we establish the asymptotic normality of the proposed estimators and propose consistent estimators of their asymptotic variances. Finite sample properties of the proposed estimators are investigated using simulation studies and the method is also illustrated with a Pima Indians diabetes data set.     

A new orthogonality-based estimation for varying-coefficient partially linear models

Varying coefficient partially linear models are usually used for longitudinal data analysis, and an interest is mainly to improve efficiency of regression coefficients. By the orthogonality estimation technology and the quadratic inference function method, we propose a new orthogonality-based estimation method to estimate parameter and nonparametric components in varying coefficient partially linear models with longitudinal data. The proposed procedure can separately estimate the parametric and nonparametric components, and the resulting estimators do not affect each other. Under some mild conditions, we establish some asymptotic properties of the resulting estimators. Furthermore, the finite sample performance of the proposed procedure is assessed by some simulation experiments.     

Semiparametric estimation for count data through weighted distributions

This paper is concerned with semiparametric discrete kernel estimators when the unknown count distribution can be considered to have a general weighted Poisson form. The estimator is constructed by multiplying the Poisson estimate with a nonparametric discrete kernel-type estimate of the Poisson weight function. Comparisons are then carried out with the ordinary discrete kernel probability mass function estimators. The Poisson weight function is thus a local multiplicative correction factor, and is considered as the uniform measure to detect departures from the equidispersed Poisson distribution. In this way, the effects of dispersion and zero-proportion with respect to the standard Poisson distribution are also minimized. This method of estimation is also applied to the weighted binomial form for the count distribution having a finite support. The proposed estimators, in addition to being simple, easy-to-implement and effective, also outperform the competing nonparametric and parametric estimators in finite-sample situations. Two examples illustrate this new semiparametric estimation.     

Least squares estimation of varying‐coefficient hazard regression with application to breast cancer dose‐intensity data

To enhance modeling flexibility, the authors propose a nonparametric hazard regression model, for which the ordinary and weighted least squares estimation and inference procedures are studied. The proposed model does not assume any parametric specifications on the covariate effects, which is suitable for exploring the nonlinear interactions between covariates, time and some exposure variable. The authors propose the local ordinary and weighted least squares estimators for the varying‐coefficient functions and establish the corresponding asymptotic normality properties. Simulation studies are conducted to empirically examine the finite‐sample performance of the new methods, and a real data example from a recent breast cancer study is used as an illustration. The Canadian Journal of Statistics 37: 659–674; 2009 © 2009 Statistical Society of Canada     

Robust designs for nearly linear regression

In this paper we seek designs and estimators which are optimal in some sense for multivariate linear regression on cubes and simplexes when the true regression function is unknown. More precisely, we assume that the unknown true regression function is the sum of a linear part plus some contamination orthogonal to the set of all linear functions in the L₂ norm with respect to Lebesgue measure. The contamination is assumed bounded in absolute value and it is shown that the usual designs for multivariate linear regression on cubes and simplices and the usual least squares estimators minimize the supremum over all possible contaminations of the expected mean square error. Additional results for extrapolation and interpolation, among other things, are discussed. For suitable loss functions optimal designs are found to have support on the extreme points of our design space.     

Bayesian mixture modeling for spatial Poisson process intensities,with applications to extreme value analysis

We propose a method for the analysis of a spatial point pattern, which is assumed to arise as a set of observations from a spatial nonhomogeneous Poisson process. The spatial point pattern is observed in a bounded region, which, for most applications, is taken to be a rectangle in the space where the process is defined. The method is based on modeling a density function, defined on this bounded region, that is directly related with the intensity function of the Poisson process. We develop a flexible nonparametric mixture model for this density using a bivariate Beta distribution for the mixture kernel and a Dirichlet process prior for the mixing distribution. Using posterior simulation methods, we obtain full inference for the intensity function and any other functional of the process that might be of interest. We discuss applications to problems where inference for clustering in the spatial point pattern is of interest. Moreover, we consider applications of the methodology to extreme value analysis problems. We illustrate the modeling approach with three previously published data sets. Two of the data sets are from forestry and consist of locations of trees. The third data set consists of extremes from the Dow Jones index over a period of 1303 days.