Semiparametric mixtures of regressions with single-index for model based clustering

In this article, we propose two classes of semiparametric mixture regression models with single-index for model based clustering. Unlike many semiparametric/nonparametric mixture regression models that can only be applied to low dimensional predictors, the new semiparametric models can easily incorporate high dimensional predictors into the nonparametric components. The proposed models are very general, and many of the recently proposed semiparametric/nonparametric mixture regression models are indeed special cases of the new models. Backfitting estimates and the corresponding modified EM algorithms are proposed to achieve optimal convergence rates for both parametric and nonparametric parts. We establish the identifiability results of the proposed two models and investigate the asymptotic properties of the proposed estimation procedures. Simulation studies are conducted to demonstrate the finite sample performance of the proposed models. Two real data applications using the new models reveal some interesting findings.

     

随机删失下半参数估计的强相合性

主要讨论了随机删失下的部分线性模型,利用基于分布函数的核估计和最小二乘法,给出了删失情况下参数和非参数部分的估计,并证明了它们的强相合性.     

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??In this paper, semiparametric estimation of a regression function in the third order partially linear autoregressive model with first order autoregressive errors is mainly studied. We suppose that the regression function has a parametric framework, and use the conditional least squares method to obtain the parameter estimators. Then semiparametric estimators of the regression function can be given by combining with the nonparametric kernel function adjustment. Furthermore, under certain conditions, the consistency of the estimators is proved. Finally, simulation research is presented to evaluate the effectiveness of the proposed method.     

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.     

Checking the adequacy of partial linear models with missing covariates at random

In this paper, we consider the goodness-of-fit for checking whether the nonparametric function in a partial linear regression model with missing covariate at random is a parametric one or not. We estimate the selection probability by using parametric and nonparametric approaches. Two score type tests are constructed with the estimated selection probability. The asymptotic distributions of the test statistics are investigated under the null and local alterative hypothesis. Simulation studies are carried out to examine the finite sample performance of the sizes and powers of the tests. We apply the proposed procedure to a data set on the AIDS clinical trial group (ACTG 315) study.     

Testing for a constant coefficient of variation in nonparametric regression by empirical processes

In the common nonparametric regression model, we consider the problem of testing the hypothesis that the coefficient of the scale and location function is constant. The test is based on a comparison of the standardized (by a local linear estimate of the scale function) observations with their mean. We show weak convergence of a centered version of this process to a Gaussian process under the null hypothesis and the alternative and use this result to construct a test for the hypothesis of a constant coefficient of variation in the nonparametric regression model. A small simulation study is also presented to investigate the finite sample properties of the new test.     

Nonparametric ANCOVA with two and three covariates

Fully nonparametric analysis of covariance with two and three covariates is considered. The approach is based on an extension of the model of Akritas et al. (Biometrika 87(3) (2000) 507). The model allows for possibly nonlinear covariate effect which can have different shape in different factor level combinations. All types of ordinal data are included in the formulation. In particular, the response distributions are not restricted to comply to any parametric or semiparametric model. In this nonparametric model, hypotheses of no main effect no interaction and no simple effect, which adjust for the covariate values, are defined through a decomposition of the conditional distribution functions of the response given to the factor level combination and covariate values. The test statistics are based on averages over the covariate values of certain Nadaraya–Watson regression quantities. Under their respective null hypotheses, such test statistics are shown to have a central χ² distribution. Small sample corrections are also provided. Simulation results and the analysis of two real datasets are also presented.     

Model checking in errors-in-variables regression

This paper discusses a class of minimum distance tests for fitting a parametric regression model to a class of regression functions in the errors-in-variables model. These tests are based on certain minimized distances between a nonparametric regression function estimator and a deconvolution kernel estimator of the conditional expectation of the parametric model being fitted. The paper establishes the asymptotic normality of the proposed test statistics under the null hypothesis and that of the corresponding minimum distance estimators. We also prove the consistency of the proposed tests against a fixed alternative and obtain the asymptotic distributions for general local alternatives. Simulation studies show that the testing procedures are quite satisfactory in the preservation of the finite sample level and in terms of a power comparison.     

A semiparametric method for estimating nonlinear autoregressive model with dependent errors

The first-order nonlinear autoregressive model is considered and a semiparametric method is proposed to estimate regression function. In the presented model, dependent errors are defined as first-order autoregressive AR(1). The conditional least squares method is used for parametric estimation and the nonparametric kernel approach is applied to estimate regression adjustment. In this case, some asymptotic behaviors and simulated results for the semiparametric method are presented. Furthermore, the method is applied for the financial data in Iran’s Tejarat-Bank.     

部分线性变系数模型改进的加权混合Profile-Liu估计

研究半参数部分线性变系数模型的有偏估计,当回归模型参数部分自变量存在多重共线性时,在随机线性约束条件下,融合Profile最小二乘估计、加权混合估计和Liu估计构造回归模型参数分量改进的加权混合Profile-Liu估计,并在一定正则条件下证明估计量的渐近性质,最后利用蒙特卡洛数值模拟验证所提出估计量的有限样本表现性.     

纵向数据半参数回归模型的估计方法

本文考虑纵向数据半参数回归模型,通过考虑纵向数据的协方差结构,基于Profile最小二乘法和局部线性拟合的方法建立了模型中参数分量、回归函数和误差方差的估计量,来提高估计的有效性,在适当条件下给出了这些估计量的相合性.并通过模拟研究将该方法与最小二乘局部线性拟合估计方法进行了比较,表明了Profile最小二乘局部线性拟合方法在有限样本情况下具有良好的性质.     

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.     

Analysis of two-sample censored data using a semiparametric mixture model

In this article we study a semiparametric mixture model for the two-sample problem with right censored data. The model implies that the densities for the continuous outcomes are related by a parametric tilt but otherwise unspecified. It provides a useful alternative to the Cox （1972） proportional hazards model for the comparison of treatments based on right censored survival data. We propose an iterative algorithm for the semiparametric maximum likelihood estimates of the parametric and nonparametric components of the model. The performance of the proposed method is studied using simulation. We illustrate our method in an application to melanoma.     

Semiparametric mixtures of nonparametric regressions

In this article, we propose and study a new class of semiparametric mixture of regression models, where the mixing proportions and variances are constants, but the component regression functions are smooth functions of a covariate. A one-step backfitting estimate and two EM-type algorithms have been proposed to achieve the optimal convergence rate for both the global parameters and the nonparametric regression functions. We derive the asymptotic property of the proposed estimates and show that both the proposed EM-type algorithms preserve the asymptotic ascent property. A generalized likelihood ratio test is proposed for semiparametric inferences. We prove that the test follows an asymptotic \(\chi ^2\)-distribution under the null hypothesis, which is independent of the nuisance parameters. A simulation study and two real data examples have been conducted to demonstrate the finite sample performance of the proposed model.     

Bayesian Semiparametric Density Estimation and Model Verification Using a Logistic–Gaussian Process

This article proposes a semiparametric model, which consists of parametric and nonparametric components, for density estimation. The parametric component represents the researcher's a priori beliefs about a likely family of density functions. The nonparametric component, which is modeled by a logistic–Gaussian process, allows the predictive distribution to deviate from the parametric family if it is inadequate. Bayesian hypothesis testing is used to examine the adequacy of the parametric model relative to the flexible alternative provided by the semiparametric model. The article presents a Markov chain Monte Carlo algorithm that efficiently handles the large number of parameters.     

半参数回归模型小波估计的弱收敛速度

本文研究了误差为鞅差序列情形下的半参数回归模型.利用小波方法,在相当一般的条件下,得到了参数、非参数估计量的弱收敛速度.     

Asymptotic efficient estimation in semiparametric nonlinear regression models

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部分线性模型基于参数信息的统计推断

针对部分线性模型提出了一种新的估计方法-Profile局部最小二乘估计,方法结合了非参数部分的参数信息.另外对于部分线性模型中非参数部分是否为某一参数函数的检验问题,基于比较原假设与备择假设下模型拟合的残差平方和的思想构造了检验统计量,并给出了计算检验p-值的精确方法和三阶矩χ2逼近方法.     

检验两个部分线性模型中非参函数相等

We propose the test statistic to check whether the nonparametric func-tions in two partially linear models are equality or not in this paper. We estimate the nonparametric function both in null hypothesis and the alternative by the local linear method, where we ignore the parametric components, and then estimate the parameters by the two stage method. The test statistic is derived, and it is shown to be asymptotically normal under the null hypothesis.     

A lack-of-fit test in Tobit errors-in-variables regression models

The problem of fitting a parametric model in Tobit errors-in-variables regression models is discussed in this paper. The proposed test is based on the supremum of the Khmaladze type transformation of a certain partial sum process of calibrated residuals. This framework covers the usual error-free Tobit model as a special case. The asymptotic null distribution of this transformed process is shown to be the same as that of a time transformed standard Brownian motion. Consistency against some fixed alternatives and asymptotic power under some local nonparametric alternatives of this test are also discussed. Simulation studies are conducted to assess the finite sample performance of the proposed test.