Two-stage local M-estimation of additive models

This paper studies local M-estimation of the nonparametric components of additive models.A two-stage local M-estimation procedure is proposed for estimating the additive components and their derivatives.Under very mild conditions,the proposed estimators of each additive component and its derivative are jointly asymptotically normal and share the same asymptotic distributions as they would be if the other components were known.The established asymptotic results also hold for two particular local M-estimations:the local least squares and least absolute deviation estimations.However,for general two-stage local M-estimation with continuous and nonlinear ψ-functions,its implementation is time-consuming.To reduce the computational burden,one-step approximations to the two-stage local M-estimators are developed.The one-step estimators are shown to achieve the same effciency as the fully iterative two-stage local M-estimators,which makes the two-stage local M-estimation more feasible in practice.The proposed estimators inherit the advantages and at the same time overcome the disadvantages of the local least-squares based smoothers.In addition,the practical implementation of the proposed estimation is considered in details.Simulations demonstrate the merits of the two-stage local M-estimation,and a real example illustrates the performance of the methodology.     

LIMITING BEHAVIOR OF RECURSIVE M-ESTIMATORS IN MULTIVARIATE LINEAR REGRESSION MODELS AND THEIR ASYMPTOTIC EFFICIENCIES

Recursive algorithms are very useful for computing M-estimators of regression coefficients and scatter parameters. In this article, it is shown that for a nondecreasing ul （t）, under some mild conditions the recursive M-estimators of regression coefficients and scatter parameters are strongly consistent and the recursive M-estimator of the regression coefficients is also asymptotically normal distributed. Furthermore, optimal recursive M-estimators, asymptotic efficiencies of recursive M-estimators and asymptotic relative efficiencies between recursive M-estimators of regression coefficients are studied.     

变系数模型变窗宽局部M-估计的相合性

用变窗宽局部M-估计对变系数模型的系数函数进行估计,得到了估计的相合性和渐近正态性.所采用的方法继承了局部多项式回归的优点并且克服了最小二乘方法缺乏稳健性的缺点.变窗宽的使用提高了局部M-估计的可塑性,并使得它们能成功地处理空间非齐性曲线、异方差性及非均匀设计密度.     

线性模型中M估计分布的随机加权方法逼近

在线性模型中,M估计的渐近分布通常都涉及到不易估计的未知误差分布的某些量,如果要估计渐近方差,就需对这些冗余参数进行估计.利用随机加权方法可以避免先对误差分布中的冗余参数进行估计.给出了当自变量是随机变量时,M估计分布的随机加权逼近,证明了M估计分布的随机加权逼近是一致相合的.当取不同的凸函数,样本大小和随机权时,进一步利用蒙特卡洛方法研究估计分布.研究表明随机权取泊松权时,不仅达到同样的效果而且可以减小计算量.     

Asymptotic Normality of the Recursive M-estimators of the Scale Parameters

In this paper, the limit distributions of the recursive M-estimators of scatter parameters in a multivariate linear model setting are studied. Under some mild conditions, the asymptotic normality of the recursive M-esimtators is established. Some Monte Carlo simulation results are presented to illustrate the performance of the recursive M-estimators.     

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In this paper, we apply the empirical likelihood technique to propose a new class of M-estimators and quantile estimators in the presence of some auxiliary information under strong mixing samples. It is shown that the proposed M-estimators and quantile estimators are consistent and asymptotically normally distributed with smaller asymptotic variances than those of the usual M-estimators and quantile estimators.     

Variable bandwidth and one-step local M-estimator

A robust version of local linear regression smoothers augmented with variable bandwidth is studied. The proposed method inherits the advantages of local polynomial regression and overcomes the shortcoming of lack of robustness of least-squares techniques. The use of variable bandwidth enhances the flexibility of the resulting local M-estimators and makes them possible to cope well with spatially inhomogeneous curves, heteroscedastic errors and nonuniform design densities. Under appropriate regularity conditions, it is shown that the proposed estimators exist and are asymptotically normal. Based on the robust estimation equation, one-step local M-estimators are introduced to reduce computational burden. It is demonstrated that the one-step local M-estimators share the same asymptotic distributions as the fully iterative M-estimators, as long as the initial estimators are good enough. In other words, the one-step local M-estimators reduce significantly the computation cost of the fully iterative M-estimators without deteriorating their performance. This fact is also illustrated via simulations.     

部分线性回归模型变窗宽一步局部M-估计

讨论了部分线性回归模型的变窗宽一步局部M-估计.用一步局部M-估计给出未知函数的估计,用平均方法给出参数估计.进一步通过两个引理证明一步M-估计的渐近正态性.所提出的方法继承了局部多项式的优点并且克服了最小二乘法缺乏稳健性的缺点.     

Limiting Behavior of M-Estimators of Regression Coefficients in High Dimensional Linear Models I. Scale Dependent Case

Asymptotics of M-estimators of the regression coefficients in linear models (both scale-variant and scale-invariant) when the number of regression coefficients tends to infinity as the sample size increases are investigated The main purpose of this study is to establish the asymptotic properties under weaker conditions than usually assumed, especially to relax the restrictions on the order of the dimension. Also, the conditions assumed and the results obtained seem easy to be extended to the multivariate linear models. In the first part of the paper, the asymptotic behavior of the ordinary (i.e., not scale-invariant) M-estimates is considered.     

Robust semiparametric M-estimation and the weighted bootstrap

M-estimation is a widely used technique for statistical inference. In this paper, we study properties of ordinary and weighted M-estimators for semiparametric models, especially when there exist parameters that cannot be estimated at the convergence rate. Results on consistency, rates of convergence for all parameters, and consistency and asymptotic normality for the Euclidean parameters are provided. These results, together with a generic paradigm for studying semiparametric M-estimators, provide a valuable extension to previous related research on semiparametric maximum-likelihood estimators (MLEs). Although penalized M-estimation does not in general fit in the framework we discuss here, it is shown for a great variety of models that many of the forgoing results still hold, including the consistency and asymptotic normality of the Euclidean parameters. For semiparametric M-estimators that are not likelihood based, general inference procedures for the Euclidean parameters have not previously been developed. We demonstrate that our paradigm leads naturally to verification of the validity of the weighted bootstrap in this setting. For illustration, several examples are investigated in detail. The new M-estimation framework and accompanying weighted bootstrap technique shed light on a universal way of investigating semiparametric models.     

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Homogeneity of variance and correlation coefficients is one of assumptions in the analysis of longitudinal data.However, the assumption can be challenged. In this paper, we mainly propose and analyze nonlinear mixed effects models for longitudinal data with exponential correlation covariance structure, intend to introduce Huber's function in the log likelihood function and get robust estimation (M-estimation) by Fisher scoring method. Score test statistics for homogeneity of variance and correlation coefficient based on M-estimation are then studied. A simulation study is carried to assess the performance of test statistics and the method we proposed in the paper is illustrated by an actual data example.     

An Approximate M—estimation for the Parameters of Mixed Regression Model

In this paper, to keep scale inveriance, we propose an approximate M-estrmation for the mixed regression model and show consistency of the estimation under weaker conditions than that in [1].     

基于M估计的线性混合模型的局部影响分析

对纵向数据的线性混合模型,用Fisher得分法得到了参数的M估计(稳健估计),给出了其渐近性质,利用影响曲率研究了M估计下的随机误差方差扰动的局部影响分析问题,并通过葡萄糖数据的实例进行了分析论证.     

用M方法检测回归系数矩阵的秩

为了确定多重线性回归模型中回归系数矩阵的秩，　本文提出了一个基于Ｍ估计的模型选择程序，　且在较弱的条件下建立了回归系数矩阵的秩的估计的强相合性。     

Asymptotic distributions of M-estimators in a spatial regression model under some fixed and stochastic spatial sampling designs

In this paper, we consider M-estimators of the regression parameter in a spatial multiple linear regression model. We establish consistency and asymptotic normality of the M-estimators when the data-sites are generated by a class of deterministic as well as a class of stochastic spatial sampling schemes. Under the deterministic sampling schemes, the data-sites are located on a regular grid but may have aninfill component. On the other hand, under the stochastic sampling schemes, locations of the data-sites are given by the realizations of a collection of independent random vectors and thus, are irregularly spaced. It is shown that scaling constants of different orders are needed for asymptotic normality under different spatial sampling schemes considered here. Further, in the stochastic case, the asymptotic covariance matrix is shown to depend on the spatial sampling density associated with the stochastic design. Results are established for M-estimators corresponding to certain non-smooth score functions including Huber’s ψ-function and the sign functions (corresponding to the sample quantiles). Research of Lahiri is partially supported by NSF grant no. DMS-0072571. Research of Mukherjee is partially supported by the Academic Research Grant R-155-000-003-112 from the National University of Singapore.     

General relative error criterion and M-estimation

Relative error rather than the error itself is of the main interest in many practical applications. Criteria based on minimizing the sum of absolute relative errors (MRE) and the sum of squared relative errors (RLS) were proposed in the different areas. Motivated by K. Chen et al.’s recent work [J. Amer. Statist. Assoc., 2010, 105: 1104-1112] on the least absolute relative error (LARE) estimation for the accelerated failure time (AFT) model, in this paper, we establish the connection between relative error estimators and the M-estimation in the linear model. This connection allows us to deduce the asymptotic properties of many relative error estimators (e.g., LARE) by the well-developed M-estimation theories. On the other hand, the asymptotic properties of some important estimators (e.g., MRE and RLS) cannot be established directly. In this paper, we propose a general relative error criterion (GREC) for estimating the unknown parameter in the AFT model. Then we develop the approaches to deal with the asymptotic normalities forM-estimators with differentiable loss functions on ? or ?\{0} in the linear model. The simulation studies are conducted to evaluate the performance of the proposed estimates for the different scenarios. Illustration with a real data example is also provided.     

Adaptive penalized M-estimation with current status data

Current status data arises when a continuous response is reduced to an indicator of whether the response is greater or less than a random threshold value. In this article we consider adaptive penalized M-estimators (including the penalized least squares estimators and the penalized maximum likelihood estimators) for nonparametric and semiparametric models with current status data, under the assumption that the unknown nonparametric parameters belong to unknown Sobolev spaces. The Cox model is used as a representative of the semiparametric models. It is shown that the modified penalized M-estimators of the nonparametric parameters can achieve adaptive convergence rates, even when the degrees of smoothing are not known in advance. consistency, asymptotic normality and inference based on the weighted bootstrap for the estimators of the regression parameter in the Cox model are also established. A simulation study is conducted for the Cox model to evaluate the finite sample efficacy of the proposed approach and to compare it with the ordinary maximum likelihood estimator. It is demonstrated that the proposed method is computationally superior.We apply the proposed approach to the California Partner Study analysis.     

Information Bounds and Efficient Estimation in a Class of Censored Transformation Models

Transformation models provide a popular tool for regression analysis of censored failure time data. The most common approach towards parameter estimation in these models is based on nonparametric profile likelihood method. Several authors proposed also ad hoc M-estimators of the Euclidean component of the model. These estimators are usually simpler to implement and many of them have good practical performance. In this paper we consider the form of the information bound for estimation of the Euclidean parameter of the model and propose a modification of the inefficient M-estimators to one-step maximum likelihood estimates.     

Estimators of scale parameters in linear regression

This note discusses the asymptotic distribution of two scale and location invariant estimators of two scale parameters in the multiple linear regression model. Both of these estimators need an initial estimator of the regression parameter vector. The asymptotic distribution of one of these estimators does not depend on this initial estimator. Both of these estimators are useful in the computation of scale and translation invariant adaptive estimators and M-estimators of the regression parameter vector.     

Sieve M-estimation for semiparametric varying-coefficient partially linear regression model

This article considers a semiparametric varying-coefficient partially linear regression model.The semiparametric varying-coefficient partially linear regression model which is a generalization of the partially linear regression model and varying-coefficient regression model that allows one to explore the possibly nonlinear effect of a certain covariate on the response variable.A sieve M-estimation method is proposed and the asymptotic properties of the proposed estimators are discussed.Our main object is to estimate the nonparametric component and the unknown parameters simultaneously.It is easier to compute and the required computation burden is much less than the existing two-stage estimation method.Furthermore,the sieve M-estimation is robust in the presence of outliers if we choose appropriate ρ( ).Under some mild conditions,the estimators are shown to be strongly consistent;the convergence rate of the estimator for the unknown nonparametric component is obtained and the estimator for the unknown parameter is shown to be asymptotically normally distributed.Numerical experiments are carried out to investigate the performance of the proposed method.