A consistent estimator for nonlinear regression models

In this paper an estimator for the general (nonlinear) regression model with random regressors is studied which is based on the Fourier transform of a certain weight function. Consistency and asymptotic normality of the estimator are established and simulation results are presented to illustrate the theoretical ones.Supported by the Hungarian National Science Foundation OTKA under Grants No. F 032060/2000 and F 046061/2004 and by the Bolyai Grant of the Hungarian Academy of Sciences.Received October 2003     

Finite sample properties for the semiparametric estimation of the intercept of a censored regression model

Financial support for this paper was provided by a C.A. Anderson Fellowship of the Cowles Foundation. I wish to thank Donald Andrews, Moshe Buchinsky, Oliver Linton, and Peter Robinson for helpful discussions. I also wish to thank three anonymous referees for their comments and suggestions. I am, of course, responsible for any remaining errors. A popular two-step estimator of the intercept of a censored regression model is compared with consistent asymptotically normal semiparametric alternatives. Using a root mean squared error criterion, the semiparametric estimators perform better for a range of bandwidth parameter choices for a variety of distributions of the errors and regressors. For error distributions that are close to the normal, however, the two-step parametric estimator performs better.     

Asymptotics and smoothing parameter selection for penalized spline regression with various loss functions

Penalized splines are used in various types of regression analyses, including non‐parametric quantile, robust and the usual mean regression. In this paper, we focus on the penalized spline estimator with general convex loss functions. By specifying the loss function, we can obtain the mean estimator, quantile estimator and robust estimator. We will first study the asymptotic properties of penalized splines. Specifically, we will show the asymptotic bias and variance as well as the asymptotic normality of the estimator. Next, we will discuss smoothing parameter selection for the minimization of the mean integrated squares error. The new smoothing parameter can be expressed uniquely using the asymptotic bias and variance of the penalized spline estimator. To validate the new smoothing parameter selection method, we will provide a simulation. The simulation results show that the consistency of the estimator with the proposed smoothing parameter selection method can be confirmed and that the proposed estimator has better behavior than the estimator with generalized approximate cross‐validation. A real data example is also addressed.     

Generalized two-sample U-statistics for clustered data

In this paper we investigate two-sample U -statistics in the case of clusters of repeated measurements observed on individuals from independent populations. The observations on the i -th individual in the first population are denoted by     , 1 ≤  i  ≤  m , and those on the k -th individual in the second population are denoted by     , 1 ≤  k  ≤  n . Given the kernel φ ( x ,  y ), we define the generalized two-sample U -statistic by
We derive the asymptotic distribution of U _{m , n} for large sample sizes. As an application we study the generalized Mann–Whitney–Wilcoxon rank sum test for clustered data.     

A consistent estimator in general functional errors-in-variables models

The problem of estimation in nonlinear functional errors-in-variables model is considered. A modified least squares estimator is studied, its consistency and asymptotic normality is established. Simulation results are also presented showing the performance of the estimator in comparison with the naive ordinary least squares estimator. Received: June 1999     

The Cox–Aalen model for recurrent‐event data with a dependent terminal event

Recurrent events with a dependent terminal event arise frequently in a wide variety of fields. In this paper, we propose a new joint model to analyze these data and model the dependence between recurrent and terminal events through shared gamma frailty. Specifically, a Cox–Aalen rate frailty model is specified for the recurrent event, and an additive hazards frailty model is specified for the terminal event. An estimating equation approach is developed for the parameters in the joint model, and the asymptotic properties of the proposed estimators are established. Simulation studies demonstrate that the proposed estimators perform well with finite samples. An application to a medical cost study of chronic heart failure patients is illustrated.     

Asymptotic results and tests for the choice of approximative models in nonlinear two-phases regression models, heteroscedatic case

This paper is devoted to the study of the least squares estimator of f for the classical, fixed design, nonlinear model X (t _i)=f(t _i)+ε(t _i), i=1,2,…,n, where the (ε(t _i))_i=1,…,n are independent second order r.v.. The estimation of f is based upon a given parametric form. In Brodeau (1993) this subject has been studied in the homoscedastic case. This time we assume that the ε(t _i) have non constant and unknown variances σ²(t _i). Our main goal is to develop two statistical tests, one for testing that f belongs to a given class of functions possibly discontinuous in their first derivative, and another for comparing two such classes. The fundamental tool is an approximation of the elements of these classes by more regular functions, which leads to asymptotic properties of estimators based on the least squares estimator of the unknown parameters. We point out that Neubauer and Zwanzig (1995) have obtained interesting results for connected subjects by using the same technique of approximation. Received: February 1996     

Diagnostics analysis for log‐Birnbaum–Saunders regression models with censored data

This article develops influence diagnostics for log‐Birnbaum–Saunders (LBS) regression models with censored data based on case‐deletion model (CDM). The one‐step approximations of the estimates in CDM are given and case‐deletion measures are obtained. Meanwhile, it is shown that CDM is equivalent to mean shift outlier model (MSOM) in LBS regression models and an outlier test is presented based on MSOM. Furthermore, we discuss a score test for homogeneity of shape parameter in LBS regression models. Two numerical examples are given to illustrate our methodology and the properties of score test statistic are investigated through Monte Carlo simulations under different censoring percentages.