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Estimation in partial linear EV models with replicated observations

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Science China Mathematics 2004年第1期47卷 144-159页

作者： cui hengjian Department of Mathematics Statistical Data Analysis Laboratory Beijing Normal University Beijing 100875China

The aim of this work is to construct the parameter estimators in the partial linear errors-in-variables (EV) models and explore their asymptotic properties. Unlike other related References, the assumption of known error covariance matrix is removed when the sample can be repeatedly drawn at each designed point from the model. The estimators of interested regression parameters, and the model error variance, as well as the nonparametric function, are constructed. Under some regular conditions, all of the estimators prove strongly consistent. Meanwhile, the asymptotic normality for the estimator of regression parameter is also presented. A simulation study is reported to illustrate our asymptotic results.

关键词： partial linear EV model, strong consistency, replicated observations.

EMPIRICAL LIKELIHOOD CONFIDENCE REGION FOR PARAMETERS IN LINEAR ERRORS-IN-VARIABLES MODELS WITH MISSING DATA

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Journal of Systems Science & Complexity 2011年第3期24卷 540-553页

作者： Juan ZHANG hengjian cui Department of Statistics Capital University of Economics and Business Beijing 100070 China. Department of Statistics and Financial MatheTnatics School of Mathematical Sciences Beijing Normal University Beijing 100875 China School of Mathematical Sciences Capital Normal University Beijing 100048 China

The multivariate linear errors-in-variables model when the regressors are missing at random in the sense of Rubin （1976） is considered in this paper. A constrained empirical likelihood confidence region for a parameter β0 in this model is proposed, which is constructed by combining the score function corresponding to the weighted squared orthogonal distance based on inverse probability with a constrained region of β0. It is shown that the empirical log-likelihood ratio at the true parameter converges to the standard chi-square distribution. Simulations show that the coverage rate of the proposed confidence region is closer to the nominal level and the length of confidence interval is narrower than those of the normal approximation of inverse probability weighted adjusted least square estimator in most cases. A real example is studied and the result supports the theory and simulation＇s conclusion.

关键词： Confidence region coverage rate empirical likelihood ratio multivariate linear errors-in- variables model weighted adjusted LS estimation.

Consistency of Chi-Squared Test with Varying Number of Classes

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Journal of Systems Science & Complexity 2015年第2期28卷 439-450页

作者： HUANG Rui cui hengjian College of Mathematic and Statistics Jishou University School of Mathematical Sciences Beijing Normal University Department of Statistics School of Mathematical SciencesCapital Normal University

The classical chi-squared goodness of fit test assumes the number of classes is fixed,meanwhile the test statistic has a limiting chi-square distribution under the null *** is well known that the number of classes varying with sample size in the test has attached more and more ***,in this situation,there is not theoretical results for the asymptotic property of such chi-squared test *** paper proves the consistency of chi-squared test with varying number of classes under some ***,the authors also give a convergence rate of KolmogorovSimirnov distance between the test statistic and corresponding chi-square distributed random *** addition,a real example and simulation results validate the reasonability of theoretical result and the superiority of chi-squared test with varying number of classes.

关键词： Consistency of chi-squared test goodness of fit test varying number of classes.

Generalized F-Test for High Dimensional Regression Coefficients of Partially Linear Models

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Journal of Systems Science & Complexity 2017年第5期30卷 1206-1226页

作者： WANG Siyang cui hengjian School of Statistics and Mathematics Central University of Finance and EconomicsBeijing 100081China School of Mathematical Sciences Capital Normal UniversityBeijing 100048China

This paper proposes a test procedure for testing the regression coefficients in high dimensional partially linear models based on the F-statistic. In the partially linear model, the authors first estimate the unknown nonlinear component by some nonparametric methods and then generalize the F-statistic to test the regression coefficients under some regular conditions. During this procedure, the estimation of the nonlinear component brings much challenge to explore the properties of generalized F-test. The authors obtain some asymptotic properties of the generalized F-test in more general cases,including the asymptotic normality and the power of this test with p/n ∈(0, 1) without normality assumption. The asymptotic result is general and by adding some constraint conditions we can obtain the similar conclusions in high dimensional linear models. Through simulation studies, the authors demonstrate good finite-sample performance of the proposed test in comparison with the theoretical results. The practical utility of our method is illustrated by a real data example.

关键词： Generalized F-test high dimensional regression partially linear models power of test.

RCV-based error density estimation in the ultrahigh dimensional additive model

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Science China Mathematics 2022年第5期65卷 1003-1028页

作者： Feng Zou hengjian cui School of Mathematical Sciences Capital Normal UniversityBeijing 100048China

In this paper,we mainly study how to estimate the error density in the ultrahigh dimensional sparse additive model,where the number of variables is larger than the sample ***,a smoothing method based on B-splines is applied to the estimation of regression ***,an improved two-stage refitted crossvalidation(RCV)procedure by random splitting technique is used to obtain the residuals of the model,and then the residual-based kernel method is applied to estimate the error density *** suitable sparse conditions,the large sample properties of the estimator,including the weak and strong consistency,as well as normality and the law of the iterated logarithm,are ***,the relationship between the sparsity and the convergence rate of the kernel density estimator is *** methodology is illustrated by simulations and a real data example,which suggests that the proposed method performs well.

关键词： ultrahigh dimensional additive model B-spline kernel density estimation refitted cross-validation method asymptotic property

Robust U-type test for high dimensional regression coefficients using refitted cross-validation variance estimation

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Science China Mathematics 2016年第12期59卷 2319-2334页

作者： GUO WenWen CHEN YongShuai cui hengjian School of Mathematical Sciences Capital Normal University

This paper aims to develop a new robust U-type test for high dimensional regression coefficients using the estimated U-statistic of order two and refitted cross-validation error variance estimation. It is proved that the limiting null distribution of the proposed new test is normal under two kinds of ordinary *** further study the local power of the proposed test and compare with other competitive tests for high dimensional data. The idea of refitted cross-validation approach is utilized to reduce the bias of sample variance in the estimation of the test statistic. Our theoretical results indicate that the proposed test can have even more substantial power gain than the test by Zhong and Chen(2011) when testing a hypothesis with outlying observations and heavy tailed distributions. We assess the finite-sample performance of the proposed test by examining its size and power via Monte Carlo studies. We also illustrate the application of the proposed test by an empirical analysis of a real data example.

关键词： high dimension regression large p small n refitted cross-validation variance estimation U-type test robust

ON WEIGHTED RANDOMLY TRIMMED MEANS

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Journal of Systems Science & Complexity 2007年第1期20卷 47-65页

作者： Ting WANG Yong LI hengjian cui School of Mathematical Sciences Statistical Data Analysis Laboratory Beijing Normal University Beijing 100875 China Institute of Information Sciences and Technology Massey University Private Bag 11222 Palmerston North New Zealand. School of Mathematical Sciences Statistical Data Analysis Laboratory Beijing Normal University Beijing 100875 China.

A class of robust location estimators called weighted randomly trimmed means are introduced and not only their consistency and asymptotic normality are proved, but their influence functions, asymptotic variances and breakdown points are also derived. They possess the same breakdown points as the median, and some of them own higher asymptotic relative efficiencies at the heavy-tailed distributions than some other well-known location estimators; whereas the trimmed means, Winsorized means and Huber＇s M-estimator possess higher asymptotic relative efficiencies at the light-tailed distributions, in which Huber＇s M-estimator is the most robust.

关键词： Asymptotic normality asymptotic relative efficiency breakdown points consistency influence function.

Robust estimation for partially linear models with large-dimensional covariates

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Science China Mathematics 2013年第10期56卷 2069-2088页

作者： ZHU LiPing LI RunZe cui hengjian School of Statistics and Management Shanghai University of Finance and Economics The Key Laboratory of Mathematical Economics (SUFE) Ministry of Education Department of Statistics and The Methodology Center The Pennsylvania State UniversityUniversity Park School of Mathematical Science Capital Normal University

We are concerned with robust estimation procedures to estimate the parameters in partially linear models with large-dimensional covariates. To enhance the interpretability, we suggest implementing a noncon- cave regularization method in the robust estimation procedure to select important covariates from the linear component. We establish the consistency for both the linear and the nonlinear components when the covariate dimension diverges at the rate of o（√n）, where n is the sample size. We show that the robust estimate of linear component performs asymptotically as well as its oracle counterpart which assumes the baseline function and the unimportant covariates were known a priori. With a consistent estimator of the linear component, we estimate the nonparametric component by a robust local linear regression. It is proved that the robust estimate of nonlinear component performs asymptotically as well as if the linear component were known in advance. Comprehensive simulation studies are carried out and an application is presented to examine the finite-sample performance of the proposed procedures.

关键词： partially linear models robust model selection smoothly clipped absolute deviation (SCAD) semiparametric models

Robust Estimation of Parameters in Nonlinear Ordinary Differential Equation Models

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Journal of Systems Science & Complexity 2016年第1期29卷 41-60页

作者： QIU Yanping HU Tao LIANG Baosheng cui hengjian School of Mathematical Sciences Beijing Normal UniversityBeijing 100875China School of Mathematical Sciences and BCMIIS Capital Normal UniversityBeijing 100048China

Ordinary differential equation(ODE) models are widely used to model dynamic processes in many scientific *** estimation is usually a challenging problem,especially in nonlinear ODE *** most popular method,nonlinear least square estimation,is shown to be strongly sensitive to *** this paper,robust estimation of parameters using M-estimators is proposed,and their asymptotic properties are obtained under some regular *** authors also provide a method to adjust Huber parameter automatically according to the ***,a method is presented to estimate the initial values of parameters and state *** efficiency and robustness are well balanced in Huber estimators,which is demonstrated via numerical simulations and chlorides data analysis.

关键词： Asymptotic properties Huber parameter ordinary differential equation robust estimation