Asymptotic properties of Lasso in high-dimensional partially linear models

We study the properties of the Lasso in the high-dimensional partially linear model where the number of variables in the linear part can be greater than the sample size. We use truncated series expansion based on polynomial splines to approximate the nonparametric component in this model. Under a sparsity assumption on the regression coefficients of the linear component and some regularity conditions, we derive the oracle inequalities for the prediction risk and the estimation error. We also provide sufficient conditions under which the Lasso estimator is selection consistent for the variables in the linear part of the model. In addition, we derive the rate of convergence of the estimator of the nonparametric function. We conduct simulation studies to evaluate the finite sample performance of variable selection and nonparametric function estimation.     

Wavelet regression estimation in nonparametric mixed effect models

We show that a nonparametric estimator of a regression function, obtained as solution of a specific regularization problem is the best linear unbiased predictor in some nonparametric mixed effect model. Since this estimator is intractable from a numerical point of view, we propose a tight approximation of it easy and fast to implement. This second estimator achieves the usual optimal rate of convergence of the mean integrated squared error over a Sobolev class both for equispaced and nonequispaced design. Numerical experiments are presented both on simulated and ERP real data.     

Stahel-Donoho kernel estimation for fixed design nonparametric regression models

This paper reports a robust kernel estimation for fixed design nonparametric regression models. A Stahel-Donoho kernel estimation is introduced, in which the weight functions depend on both the depths of data and the distances between the design points and the estimation points. Based on a local approximation, a computational technique is given to approximate to the incomputable depths of the errors. As a result the new estimator is computationally efficient. The proposed estimator attains a high breakdown point and has perfect asymptotic behaviors such as the asymptotic normality and convergence in the mean squared error. Unlike the depth-weighted estimator for parametric regression models, this depth-weighted nonparametric estimator has a simple variance structure and then we can compare its efficiency with the original one. Some simulations show that the new method can smooth the regression estimation and achieve some desirable balances between robustness and efficiency.     

A Test for Additivity in Nonparametric Regression

A simple consistent test of additivity in a multiple nonparametric regression model is proposed, where data are observed on a lattice. The new test is based on an estimator of the L ²-distance between the (unknown) nonparametric regression function and its best approximation by an additive nonparametric regression model. The corresponding test-statistic is the difference of a classical ANOVA style statistic in a two-way layout with one observation per cell and a variance estimator in a homoscedastic nonparametric regression model. Under the null hypothesis of additivity asymptotic normality is established with a limiting variance which involves only the variance of the error of measurements. The results are extended to models with an approximate lattice structure, a heteroscedastic error structure and the finite sample behaviour of the proposed procedure is investigated by means of a simulation study.     

Censored multiple regression by the method of average derivatives

This paper proposes a technique [termed censored average derivative estimation (CADE)] for studying estimation of the unknown regression function in nonparametric censored regression models with randomly censored samples. The CADE procedure involves three stages: firstly-transform the censored data into synthetic data or pseudo-responses using the inverse probability censoring weighted (IPCW) technique, secondly estimate the average derivatives of the regression function, and finally approximate the unknown regression function by an estimator of univariate regression using techniques for one-dimensional nonparametric censored regression. The CADE provides an easily implemented methodology for modelling the association between the response and a set of predictor variables when data are randomly censored. It also provides a technique for “dimension reduction” in nonparametric censored regression models. The average derivative estimator is shown to be root-n consistent and asymptotically normal. The estimator of the unknown regression function is a local linear kernel regression estimator and is shown to converge at the optimal one-dimensional nonparametric rate. Monte Carlo experiments show that the proposed estimators work quite well.     

A comparison of kriging with nonparametric regression methods

“Kriging” is the name of a parametric regression method used by hydrologists and mining engineers, among others. Features of the kriging approach are that it also provides an error estimate and that it can conveniently be employed also to estimate the integral of the regression function. In the present work, the kriging method is described and some of its statistical characteristics are explored. Also, some extensions of the nonparametric regression approach are made so that it too displays the kriging features. In particular, a “data driven” estimator of the expected square error is derived. Theoretical and computational comparisons of the kriging and nonparametric regressors are offered.     

Robust Depth-Weighted Wavelet for Nonparametric Regression Models

In the nonparametric regression models, the original regression estimators including kernel estimator, Fourier series estimator and wavelet estimator are always constructed by the weighted sum of data, and the weights depend only on the distance between the design points and estimation points. As a result these estimators are not robust to the perturbations in data. In order to avoid this problem, a new nonparametric regression model, called the depth-weighted regression model, is introduced and then the depth-weighted wavelet estimation is defined. The new estimation is robust to the perturbations in data, which attains very high breakdown value close to 1/2. On the other hand, some asymptotic behaviours such as asymptotic normality are obtained. Some simulations illustrate that the proposed wavelet estimator is more robust than the original wavelet estimator and, as a price to pay for the robustness, the new method is slightly less efficient than the original method.     

Hazard function estimation with cause-of-death data missing at random

Hazard function estimation is an important part of survival analysis. Interest often centers on estimating the hazard function associated with a particular cause of death. We propose three nonparametric kernel estimators for the hazard function, all of which are appropriate when death times are subject to random censorship and censoring indicators can be missing at random. Specifically, we present a regression surrogate estimator, an imputation estimator, and an inverse probability weighted estimator. All three estimators are uniformly strongly consistent and asymptotically normal. We derive asymptotic representations of the mean squared error and the mean integrated squared error for these estimators and we discuss a data-driven bandwidth selection method. A simulation study, conducted to assess finite sample behavior, demonstrates that the proposed hazard estimators perform relatively well. We illustrate our methods with an analysis of some vascular disease data.     

Semiparametric estimation of average treatment effect through a random coefficient dummy endogenous variable model

This paper provides an estimation procedure for average treatment effect through a random coefficient dummy endogenous variable model. A leading example of the model is estimating the effect of a training program on earnings. The model is composed of two equations: an outcome equation and a decision equation. Given the linear restriction in outcome and decision equations, Chen (1999) provided a distribution-free estimation procedure under conditional symmetric error distributions. In this paper we extend Chen’s estimator by relaxing the linear index into a nonparametric function, which greatly reduces the risk of model misspecification. A two-step approach is proposed: the first step uses a nonparametric regression estimator for the decision variable, and the second step uses an instrumental variables approach to estimate average treatment effect in the outcome equation. The proposed estimator is shown to be consistent and asymptotically normally distributed. Furthermore, we investigate the finite performance of our estimator by a Monte Carlo study and also use our estimator to study the return of college education in different periods of China. The estimates seem more reasonable than those of other commonly used estimators.     

纵向数据下半参数回归模型的统计分析

对于纵向数据下半参数回归模型,基于广义估计方程和一般权函数方法构造了模型中参数分量和非参数分量的估计.在适当的条件下证明了参数估计量具有渐近正态性,并得到了非参数回归函数估计量的最优收敛速度.通过模拟研究说明了所提出的估计量在有限样本下的精确性.     

连续时间下非参数回归模型的误差密度估计

本文研究连续时间下非参数回归的误差密度估计问题，给出误差密度的一个核估计量，利用回归函数的核估计在紧区间上一致均收敛的结论证明了该统计量渐近无偏差，均方相合法，并说明了该核估计中窗宽选取的办法。     

Nonparametric regression estimation with general parametric error covariance

The asymptotic distribution for the local linear estimator in nonparametric regression models is established under a general parametric error covariance with dependent and heterogeneously distributed regressors. A two-step estimation procedure that incorporates the parametric information in the error covariance matrix is proposed. Sufficient conditions for its asymptotic normality are given and its efficiency relative to the local linear estimator is established. We give examples of how our results are useful in some recently studied regression models. A Monte Carlo study confirms the asymptotic theory predictions and compares our estimator with some recently proposed alternative estimation procedures.     

Estimation of the slope in a linear functional relationship

Summary Two nonparametric estimators of the slope of a regression line with error on both variables are considered, each of them being defined as the zero-crossing of a stochastic process whose sample paths are monotone. Their asymptotic behaviour is derived from the local asymptotic behaviour of the underlying processes. One of the estimators is a nonparametric version of Wald's (1940) estimator.This research was supported by the Swiss National Science Foundation     

Modal Regression Based on Nonparametric Quantile Estimator

Modal regression based on nonparametric quantile estimator is given. Unlike the traditional mean and median regression, modal regression uses mode but not mean or median to represent the center of a conditional distribution, which helps the model to be more robust for outliers, asymmetric or heavy-taileddistribution. Most of solutions for modal regression are based on kernel estimation of density. This paper studies a new solution for modal regression by means of nonparametric quantile estimator. This method builds on the fact that the distribution function is the inverse of the quantile function, then the flexibility of nonparametric quantile estimator is utilized to improve the estimation of modal function. The simulations and application show that the new model outperforms the modal regression model via linear quantile function estimation.     

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Berkson测量误差模型在工业、农业、流行病学、经济学 等领域有广泛的应用. 本文考虑了一类多元超结构Berkson测量误差模型, 给出了该模型中参数的相合估计, 推导了估计的渐近分布, 并把该方法应用到一元超结构Berkson测量误差模型中, 最后给出了模拟结果和实例分析来说明本文所提出的估计方法的表现.     

Complete f-moment Convergence for Widely Orthant Dependent Random Variables and Its Application in Nonparametric Models

In this paper, we study the complete f-moment convergence for widely orthant dependent (WOD, for short) random variables. A general result on complete f-moment convergence for arrays of rowwise WOD random variables is obtained. As applications, we present some new results on complete f-moment convergence for WOD random variables. We also give an application to nonparametric regression models based on WOD errors by using the complete convergence that we established. Finally, the choice of the fixed design points and the weight functions for the nearest neighbor estimator are proposed, and a numerical simulation is provided to verify the validity of the theoretical result.     

Estimating and Visualizing Conditional Densities

Abstract

We consider the kernel estimator of conditional density and derive its asymptotic bias, variance, and mean-square error. Optimal bandwidths (with respect to integrated mean-square error) are found and it is shown that the convergence rate of the density estimator is order n ^–2/3. We also note that the conditional mean function obtained from the estimator is equivalent to a kernel smoother. Given the undesirable bias properties of kernel smoothers, we seek a modified conditional density estimator that has mean equivalent to some other nonparametric regression smoother with better bias properties. It is also shown that our modified estimator has smaller mean square error than the standard estimator in some commonly occurring situations. Finally, three graphical methods for visualizing conditional density estimators are discussed and applied to a data set consisting of maximum daily temperatures in Melbourne, Australia.     

Nonparametric Regression with Singular Design

Theories of nonparametric regression are usually based on the assumption that the design density exists. However, in some applications such as those involving high-dimensional or chaotic time series data, the design measure may be singular and may be likely to have a fractal (nonintegral) dimension. In this paper, the popular Nadaraya–Watson estimator is studied under the general setup that the continuity of the design measure is governed by the local or pointwise dimension. It will be shown in the iid setup that the nonparametric regression estimator achieves a convergence rate which is dependent only on the pointwise dimension. The case of time series data is also studied. For the latter case, a new mixing condition is introduced, and an assumption of marginal or joint density is completely avoided. Three examples, a fractal regression and two applications for predicting chaotic time series, are used to illustrate the implications of the obtained results.     

Estimating the error distribution in nonparametric multiple regression with applications to model testing

In this paper we consider the estimation of the error distribution in a heteroscedastic nonparametric regression model with multivariate covariates. As estimator we consider the empirical distribution function of residuals, which are obtained from multivariate local polynomial fits of the regression and variance functions, respectively. Weak convergence of the empirical residual process to a Gaussian process is proved. We also consider various applications for testing model assumptions in nonparametric multiple regression. The model tests obtained are able to detect local alternatives that converge to zero at an n^−1/2-rate, independent of the covariate dimension. We consider in detail a test for additivity of the regression function.