M-Estimation for partially functional linear regression model based on splines

ABSTRACT

M-estimation is a widely used technique for robust statistical inference. In this paper, we study robust partially functional linear regression model in which a scale response variable is explained by a function-valued variable and a finite number of real-valued variables. For the estimation of the regression parameters, which include the infinite dimensional function as well as the slope parameters for the real-valued variables, we use polynomial splines to approximate the slop parameter. The estimation procedure is easy to implement, and it is resistant to heavy-tailederrors or outliers in the response. The asymptotic properties of the proposed estimators are established. Finally, we assess the finite sample performance of the proposed method by Monte Carlo simulation studies.     

Robust estimation for partially linear single-index model

In this article, we investigate a new estimation approach for the partially linear single-index model based on modal regression method, where the non parametric function is estimated by penalized spline method. Moreover, we develop an expection maximum (EM)-type algorithm and establish the large sample properties of the proposed estimation method. A distinguishing characteristic of the newly proposed estimation is robust against outliers through introducing an additional tuning parameter which can be automatically selected using the observed data. Simulation studies and real data example are used to evaluate the finite-sample performance, and the results show that the newly proposed method works very well.     

S-estimator in partially linear regression models

In this paper, a robust estimator is proposed for partially linear regression models. We first estimate the nonparametric component using the penalized regression spline, then we construct an estimator of parametric component by using robust S-estimator. We propose an iterative algorithm to solve the proposed optimization problem, and introduce a robust generalized cross-validation to select the penalized parameter. Simulation studies and a real data analysis illustrate that the our proposed method is robust against outliers in the dataset or errors with heavy tails.     

Statistical estimation for a partially linear single-index model with errors in all variables

This article considers partially linear single-index models with errors in all variables. By using the Pseudo ? θ method (Liang, Härdle, and Carroll 1999), local linear regression and simulation-extrapolation (SIMEX) technique (Cook and Stefanski 1994), we propose an efficient methodology to estimate the current model. Under certain conditions the asymptotic properties of proposed estimators are obtained. Some simulation experiments and an application are conducted to illustrate our proposed method.     

The strong consistency of M-estimates in linear models with extended negatively dependent errors

In this paper, we first establish the strong convergence for weighted sums of extended negatively dependent (END) random variables. Based on the strong convergence and Bernstein inequality, we obtain the strong consistency of M-estimates of the regression parameters in a linear model for END random errors under some mild moment conditions. The results generalize and improve the ones obtained in the literature to the case of END random errors.     

Estimation for the censored partially linear quantile regression models

In this article, we develop estimation procedures for partially linear quantile regression models, where some of the responses are censored by another random variable. The nonparametric function is estimated by basis function approximations. The estimation procedure is easy to implement through existing weighted quantile regression, and it requires no specification of the error distributions. We show the large-sample properties of the resulting estimates, the proposed estimator of the regression parameter is root-n consistent and asymptotically normal and the estimator of the functional component achieves the optimal convergence rate of the nonparametric function. The proposed method is studied via simulations and illustrated with the analysis of a primary biliary cirrhosis (BPC) data.     

Statistical inference on asymptotic properties of two estimators for the partially linear single-index models

The outer product of gradients (OPG) estimation procedure based on least squares (LS) approach has been presented by Xia et al. [An adaptive estimation of dimension reduction space. J Roy Statist Soc Ser B. 2002;64:363–410] to estimate the single-index parameter in partially linear single-index models (PLSIM). However, its asymptotic property has not been established yet and the efficiency of LS-based method can be significantly affected by outliers and heavy-tailed distributions. In this paper, we firstly derive the asymptotic property of OPG estimator developed by Xia et al. [An adaptive estimation of dimension reduction space. J Roy Statist Soc Ser B. 2002;64:363–410] in theory, and a novel robust estimation procedure combining the ideas of OPG and local rank (LR) inference is further developed for PLSIM along with its theoretical property. Then, we theoretically derive the asymptotic relative efficiency (ARE) of the proposed LR-based procedure with respect to LS-based method, which is shown to possess an expression that is closely related to that of the signed-rank Wilcoxon test in comparison with the t-test. Moreover, we demonstrate that the new proposed estimator has a great efficiency gain across a wide spectrum of non-normal error distributions and almost not lose any efficiency for the normal error. Even in the worst case scenarios, the ARE owns a lower bound equalling to 0.864 for estimating the single-index parameter and a lower bound being 0.8896 for estimating the nonparametric function respectively, versus the LS-based estimators. Finally, some Monte Carlo simulations and a real data analysis are conducted to illustrate the finite sample performance of the estimators.     

Asymptotic properties for estimates of nonparametric regression model with martingale difference errors

Consider the nonparametric regression model with martingale difference errors. Nonparametric estimator g _n (x) of regression function g(x) will be introduced, and its asymptotic properties are studied. In particular, the pointwise and uniform convergence of g _n (x) and its asymptotic normality will be investigated. This extends the earlier work on independent random errors.     

Asymptotics of M-estimator in multivariate linear regression models for a class of random errors

It is known that linear regression models have immense applications in various areas such as engineering technology, economics and social sciences. In this paper, we investigate the asymptotic properties of M-estimator in multivariate linear regression model based on a class of random errors satisfying a generalised Bernstein-type inequality. By using the generalised Bernstein-type inequality, we obtain a general result on almost sure convergence for a class of random variables and then obtain the strong consistency for the M-estimator in multivariate linear regression models under some mild conditions. The result extends or improves some existing ones in the literature. Moreover, we also consider the case when the dimension $p$ tends to infinity by establishing the rate of almost sure convergence for a class of random variables satisfying generalised Bernstein-type inequality. Some numerical simulations are also provided to verify the validity of the theoretical results.     

Testing for the parametric component of partially linear EV models under random censorship

This paper investigates the hypothesis test of the parametric component in partially linear errors-in-variables (EV) model with random censorship. We construct two test statistics based on the difference of the corrected residual sum of squares and empirical likelihood ratio under the null and alternative hypotheses. It is shown that the limiting distributions of the proposed test statistics are both weighted sum of independent standard chi-squared distribution with one degree of freedom under the null hypothesis. Based on the adjusted test statistics, we further develop two new types of test procedures. Finite sample performance of the proposed test procedures is evaluated by extensive simulation studies.     

Estimation for generalized partially functional linear additive regression model

In practice, it is not uncommon to encounter the situation that a discrete response is related to both a functional random variable and multiple real-value random variables whose impact on the response is nonlinear. In this paper, we consider the generalized partial functional linear additive models (GPFLAM) and present the estimation procedure. In GPFLAM, the nonparametric functions are approximated by polynomial splines and the infinite slope function is estimated based on the principal component basis function approximations. We obtain the estimator by maximizing the quasi-likelihood function. We investigate the finite sample properties of the estimation procedure via Monte Carlo simulation studies and illustrate our proposed model by a real data analysis.     

Complete consistency for the estimator of nonparametric regression model based on martingale difference errors

Abstract

In this paper, we study the complete consistency for the estimator of nonparametric regression model based on martingale difference errors, and obtain the convergence rates of the complete consistency by using the inequalities for martingale difference sequence. Finally, some simulations are illustrated.     

Weak consistency of M-estimator in linear regression model with asymptotically almost negatively associated errors

Abstract

This paper studies a linear regression model with asymptotically almost negatively associated (AANA, in short) random errors. Under some mild conditions, the weak consistency of M-estimator of the unknown parameter is investigated, which extend the corresponding results for independent random errors and negatively associated (NA, in short) random errors. At last, two simulation examples are presented to verify the weak consistency of M-estimator in the model.     

Ridge regression estimators in the linear regression models with non-spherical errors

The present paper considers a family of ordinary ridge regression estimators in the linear regression model when the disturbances covariance matrix depends upon a few unknown parameters. An asymptotic expansion for the distribution of the ridge regression estimator is developed and under the quadratic loss function its asymptotic risk is compared with that of the feasible GLS estimator.     

Asymptotic distribution of least square estimators for linear models with dependent errors

In this paper, we consider the usual linear regression model in the case where the error process is assumed strictly stationary. We use a result from Hannan (Central limit theorems for time series regression. Probab Theory Relat Fields. 1973;26(2):157–170), who proved a Central Limit Theorem for the usual least squares estimator under general conditions on the design and on the error process. Whatever the design satisfying Hannan's conditions, we define an estimator of the covariance matrix and we prove its consistency under very mild conditions. As an application, we show how to modify the usual tests on the linear model in this dependent context, in such a way that the type-I error rate remains asymptotically correct, and we illustrate the performance of this procedure through different sets of simulations.     

B-spline estimation for partially linear varying coefficient composite quantile regression models

Abstract

In this article, a new composite quantile regression estimation (CQR) approach is proposed for partially linear varying coefficient models (PLVCM) under composite quantile loss function with B-spline approximations. The major advantage of the proposed procedures over the existing ones is easy to implement using existing software, and it requires no specification of the error distributions. Under the regularity conditions, the consistency and asymptotic normality of the estimators are also derived. Finally, a simulation study and a real data application are undertaken to assess the finite sample performance of the proposed estimation procedure.     

General linear estimators under the prediction error sum of squares criterion in a linear regression model

In this paper, the notion of the general linear estimator and its modified version are introduced using the singular value decomposition theorem in the linear regression model y=X β+e to improve some classical linear estimators. The optimal selections of the biasing parameters involved are theoretically given under the prediction error sum of squares criterion. A numerical example and a simulation study are finally conducted to illustrate the superiority of the proposed estimators.     

Consistency of maximum likelihood estimators for the regime-switching GARCH model

The regime-switching GARCH (generalized autoregressive conditionally heteroscedastic) model incorporates the idea of Markov switching into the more restrictive GARCH model, which significantly extends the GARCH model. However, the statistical inference for such an extended model is rather difficult because observations at any time point then depend on the whole regime path and the likelihood becomes intractable quickly as the length of observations increases. In this paper, by transforming it into an infinite order ARCH model, we obtain the possibility of writing a likelihood which can be handled directly and the consistency of the maximum likelihood estimators is proved. Simulation studies to illustrate the consistency and asymptotic normality of the estimators (for both Gaussian and non-Gaussian innovations) and a model specification problem are presented.     

Weighted composite quantile regression for partially linear varying coefficient models

Partially linear varying coefficient models (PLVCMs) with heteroscedasticity are considered in this article. Based on composite quantile regression, we develop a weighted composite quantile regression (WCQR) to estimate the non parametric varying coefficient functions and the parametric regression coefficients. The WCQR is augmented using a data-driven weighting scheme. Moreover, the asymptotic normality of proposed estimators for both the parametric and non parametric parts are studied explicitly. In addition, by comparing the asymptotic relative efficiency theoretically and numerically, WCQR method all outperforms the CQR method and some other estimate methods. To achieve sparsity with high-dimensional covariates, we develop a variable selection procedure to select significant parametric components for the PLVCM and prove the method possessing the oracle property. Both simulations and data analysis are conducted to illustrate the finite-sample performance of the proposed methods.