带有随机设计的非参数回归模型的异方差小波检验

该文讨论了带有随机设计的非参数回归模型的异方差小波检验. 首先给出了回归模型的条件方差函数的经验小波系数, 然后证明了它们是渐近独立和正态的. 基于 Fan (1996) 的方法, 构造了异方差检验统计量.最后通过数值模拟,作者检验了该文所提出的方法的有效性.模拟结果表明该文所提出的检验方法在水平和功效方面表现良好.     

Testing heteroscedasticity in nonparametric regression models based on residual analysis

The importance of detecting heteroscedasticity in regression analysis is widely recognized because efficient inference for the regression function requires that heteroscedasticity should be taken into account. In this paper, a simple test for heteroscedasticity is proposed in nonparametric regression based on residual analysis. Furthermore, some simulations with a comparison with Dette and Munk＇s method are conducted to evaluate the performance of the proposed test. The results demonstrate that the method in this paper performs quite satisfactorily and is much more powerful than Dette and Munk＇s method in some cases.     

Pairwise distance-based heteroscedasticity test for regressions

Science China Mathematics - In this study, we propose nonparametric testing for heteroscedasticity in nonlinear regression models based on pairwise distances between points in a sample. The test...     

Heteroscedasticity checks for regression models

For checking on heteroscedasticity in regression models, a unified approach is proposed to constructing test statistics in parametric and nonparametric regression models. For nonparametric regression, the test is not affected sensitively by the choice of smoothing parameters which are involved in estimation of the nonparametric regression function. The limiting null distribution of the test statistic remains the same in a wide range of the smoothing parameters. When the covariate is one-dimensional, the tests are, under some conditions, asymptotically distribution-free. In the high-dimensional cases, the validity of bootstrap approximations is investigated. It is shown that a variant of the wild bootstrap is consistent while the classical bootstrap is not in the general case, but is applicable if some extra assumption on conditional variance of the squared error is imposed. A simulation study is performed to provide evidence of how the tests work and compare with tests that have appeared in the literature. The approach may readily be extended to handle partial linear, and linear autoregressive models.     

随机权函数半参数回归模型的导方差统计分析

在回归分析中,观测值的方差齐性只是一个基本的假定,在参数、半参数和非参数回归模型中关于异方差检验和估计问题已有很多研究.本文在冉昊和朱忠义(2004)讨论的半参数回归模型的基础上,用随机参数方法,讨论随机权函数半参数回归模型中的异方差检验问题,得到了方差齐性检验Score统计量,同时,当半参数模型存在异方差时,本文还给出了估计方差的方法.     

Statistical inference of partially linear regression models with heteroscedastic errors

The authors study a heteroscedastic partially linear regression model and develop an inferential procedure for it. This includes a test of heteroscedasticity, a two-step estimator of the heteroscedastic variance function, semiparametric generalized least-squares estimators of the parametric and nonparametric components of the model, and a bootstrap goodness of fit test to see whether the nonparametric component can be parametrized.     

Two tests for heteroscedasticity in nonparametric regression

In this paper, two new tests for heteroscedasticity in nonparametric regression are presented and compared. The first of these tests consists in first estimating nonparametrically the unknown conditional variance function and then using a classical least-squares test for a general linear model to test whether this function is a constant. The second test is based on using an overall distance between a nonparametric estimator of the conditional variance function and a parametric estimator of the variance of the model under the assumption of homoscedasticity. A bootstrap algorithm is used to approximate the distribution of this test statistic. Extended versions of both procedures in two directions, first, in the context of dependent data, and second, in the case of testing if the variance function is a polynomial of a certain degree, are also described. A broad simulation study is carried out to illustrate the finite sample performance of both tests when the observations are independent and when they are dependent.     

Heteroscedasticity check in nonlinear semiparametric models based on nonparametric variance function

The assumption of homoscedasticity has received much attention in classical analysis of regression. Heteroscedasticity tests have been well studied in parametric and nonparametric regressions. The aim of this paper is to present a test of heteroscedasticity for nonlinear semiparametric regression models with nonparametric variance function. The validity of the proposed test is illustrated by two simulated examples and a real data example.     

随机设计下非参数回归模型变点的小波检测与估计

A wavelet method of detection and estimation of change points in nonparametric regression models under random design is proposed. The confidence bound of our test is derived by using the test statistics based on empirical wavelet coefficients as obtained by wavelet transformation of the data which is observed with noise. Moreover, the consistence of the test is proved while the rate of convergence is given. The method turns out to be effective after being tested on simulated examples and applied to IBM stock market data.     

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.     

On spline approximation of sliced inverse regression

The dimension reduction is helpful and often necessary in exploring the nonparametric regression structure.In this area,Sliced inverse regression (SIR) is a promising tool to estimate the central dimension reduction (CDR) space.To estimate the kernel matrix of the SIR,we herein suggest the spline approximation using the least squares regression.The heteroscedasticity can be incorporated well by introducing an appropriate weight function.The root-n asymptotic normality can be achieved for a wide range choice of knots.This is essentially analogous to the kernel estimation.Moreover, we also propose a modified Bayes information criterion (BIC) based on the eigenvalues of the SIR matrix.This modified BIC can be applied to any form of the SIR and other related methods.The methodology and some of the practical issues are illustrated through the horse mussel data.Empirical studies evidence the performance of our proposed spline approximation by comparison of the existing estimators.     

Test d'hétéroscédasticité quand les covariables sont fonctionnelles

We present in this paper a consistent nonparametric test for heteroscedasticity when data are of functional kind. The latter is constructed by evaluating the difference between the conditional and unconditional variances. We show the asymptotic normality of the statistical test under the null hypothesis. In addition, we prove that this test is consistent against all deviations from homoscedasticity condition.     

Statistical inference for panel data semiparametric partially linear regression models with heteroscedastic errors

We consider a panel data semiparametric partially linear regression model with an unknown parameter vector for the linear parametric component, an unknown nonparametric function for the nonlinear component, and a one-way error component structure which allows unequal error variances (referred to as heteroscedasticity). We develop procedures to detect heteroscedasticity and one-way error component structure, and propose a weighted semiparametric least squares estimator (WSLSE) of the parametric component in the presence of heteroscedasticity and/or one-way error component structure. This WSLSE is asymptotically more efficient than the usual semiparametric least squares estimator considered in the literature. The asymptotic properties of the WSLSE are derived. The nonparametric component of the model is estimated by the local polynomial method. Some simulations are conducted to demonstrate the finite sample performances of the proposed testing and estimation procedures. An example of application on a set of panel data of medical expenditures in Australia is also illustrated.     

小波检测并估计非参函数变点

研究随机设计下非参函数变点的小波检测与估计问题.将小波方法与设计点转化方法相结合给出变点的检测统计量并研究检测的一致性.给出了变点个数和变点位置的估计量,证明了变点个数估计量的相合性并得到变点位置估计量的收敛速度.     

完全与删失数据下回归函数小波估计的强一致收敛速度

在完全和右删失数据下，构造了回归函数g(x)的小波估计和改良小波估计，得到了估计量的若干强一致收敛速度。     

Fast Nonparametric Quantile Regression With Arbitrary Smoothing Methods

The calculation of nonparametric quantile regression curve estimates is often computationally intensive, as typically an expensive nonlinear optimization problem is involved. This article proposes a fast and easy-to-implement method for computing such estimates. The main idea is to approximate the costly nonlinear optimization by a sequence of well-studied penalized least squares-type nonparametric mean regression estimation problems. The new method can be paired with different nonparametric smoothing methods and can also be applied to higher dimensional settings. Therefore, it provides a unified framework for computing different types of nonparametric quantile regression estimates, and it also greatly broadens the scope of the applicability of quantile regression methodology. This wide applicability and the practical performance of the proposed method are illustrated with smoothing spline and wavelet curve estimators, for both uni- and bivariate settings. Results from numerical experiments suggest that estimates obtained from the proposed method are superior to many competitors. This article has supplementary material online.     

Adaptive nonparametric regression on spin fiber bundles

The construction of adaptive nonparametric procedures by means of wavelet thresholding techniques is now a classical topic in modern mathematical statistics. In this paper, we extend this framework to the analysis of nonparametric regression on sections of spin fiber bundles defined on the sphere. This can be viewed as a regression problem where the function to be estimated takes as its values algebraic curves (for instance, ellipses) rather than scalars, as usual. The problem is motivated by many important astrophysical applications, concerning, for instance, the analysis of the weak gravitational lensing effect, i.e. the distortion effect of gravity on the images of distant galaxies. We propose a thresholding procedure based upon the (mixed) spin needlets construction recently advocated by Geller and Marinucci (2008, 2010) and Geller et al. (2008, 2009), and we investigate their rates of convergence and their adaptive properties over spin Besov balls.     

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.     

DIMENSION-REDUCTION TYPE TESTFOR LINEARITY OF ASTOCHASTIC REGRESSION MODEL

1.IntroductionLinearregressionmodelsarewidelyusedinstatisticalanalysisofexperimentalandobservationaldata,thatis,oneoftenemploysastandardlinearmodely=or K: E,a.s.,(1.1)todostatisticalanalysis,whereydenotesascalaroutcomevariableand2denotesaP-dimensionalcolumnvectorofregressorvariables.Thismodelmeansthattheprojectionofthepdimensionalexplanatory2ontotheone-dimensionalsubspaceadZcapturesalltheinformationweneedtoknowabouttheoutcomevariabley.Thisisadimension-reductionmodel.Hencewemayreachthegoalofd…     

回归模型的同方差检验

本文利用局部经验似然和WNW方法对条件分布函数和条件分位数进行估计,并利用条件分位数的方法对回归模型中的误差方差进行了同方差假设检验,获得了零假设下检验统计量的渐近分布为X2分布．模拟计算表明同方差假设检验的条件分位数方法具有较好的功效．