Efficient estimation for semiparametric varying- coefficient partially linear regression models with current status data

This article considers a semiparametric varying-coefficient partially linear regression model with current status data. 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 maximum likelihood estimation method is proposed and the asymptotic properties of the proposed estimators are discussed. Under some mild conditions, the estimators are shown to be strongly consistent. The convergence rate of the estimator for the unknown smooth function is obtained and the estimator for the unknown parameter is shown to be asymptotically efficient and normally distributed. Simulation studies are conducted to examine the small-sample properties of the proposed estimates and a real dataset is used to illustrate our approach.     

Robust Mean-Squared Error Estimation of Multiple Signals in Linear Systems Affected by Model and Noise Uncertainties

This paper is a continuation of the work in [11] and [2] on the problem of estimating by a linear estimator, N unobservable input vectors, undergoing the same linear transformation, from noise-corrupted observable output vectors. Whereas in the aforementioned papers, only the matrix representing the linear transformation was assumed uncertain, here we are concerned with the case in which the second order statistics of the noise vectors (i.e., their covariance matrices) are also subjected to uncertainty. We seek a robust mean-squared error estimator immuned against both sources of uncertainty. We show that the optimal robust mean-squared error estimator has a special form represented by an elementary block circulant matrix, and moreover when the uncertainty sets are ellipsoidal-like, the problem of finding the optimal estimator matrix can be reduced to solving an explicit semidefinite programming problem, whose size is independent of N. The research was partially supported by BSF grant #2002038     

Variable Selection for Partially Linear Models via Adaptive LASSO

Partially linear model is a class of commonly used semiparametric models, this paper focus on variable selection and parameter estimation for partially linear models via adaptive LASSO method. Firstly, based on profile least squares and adaptive LASSO method, the adaptive LASSO estimator for partially linear models are constructed, and the selections of penalty parameter and bandwidth are discussed. Under some regular conditions, the consistency and asymptotic normality for the estimator are investigated, and it is proved that the adaptive LASSO estimator has the oracle properties. The proposed method can be easily implemented. Finally a Monte Carlo simulation study is conducted to assess the finite sample performance of the proposed variable selection procedure, results show the adaptive LASSO estimator behaves well.     

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.     

Efficient estimation of a varying-coefficient partially linear binary regression model

This article considers a semiparametric varying-coefficient partially linear binary regression model. The semiparametric varying-coefficient partially linear regression binary model which is a generalization of binary 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 maximum likelihood estimation method is proposed and the asymptotic properties of the proposed estimators are discussed. One of our main objects is to estimate nonparametric component and the unknowen parameters simultaneously. It is easier to compute, and the required computation burden is much less than that of the existing two-stage estimation method. Under some mild conditions, the estimators are shown to be strongly consistent. The convergence rate of the estimator for the unknown smooth function is obtained, and the estimator for the unknown parameter is shown to be asymptotically efficient and normally distributed. Simulation studies are carried out to investigate the performance of the proposed method.     

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This paper concerns with the estimation of a fixed effects panel data partially linear regression model with the idiosyncratic errors being an autoregressive process. For fixed effects short time series panel data, the commonly used autoregressive error structure fitting method will not result in a consistent estimator of the autoregressive coefficients. Here we propose an alternative estimation and show that the resulting estimator of the autoregressive coefficients is consistent and this method is workable for any order autoregressive error structure. Moreover, combining the B-spline approximation, profile least squares dummy variable (PLSDV) technique and consistently estimated the autoregressive error structure, we develop a weighted PLSDV estimator for the parametric component and a weighted B-spline series (BS) estimator for the nonparametric component. The weighted PLSDV estimator is shown to be asymptotically normal and more asymptotically efficient than the one which ignores the error autoregressive structure. In addition, this paper derives the asymptotic bias of the weighted BS estimator and establish its asymptotic normality as well. Simulation studies and an example of application are conducted to illustrate the finite sample performance of the proposed procedures.     

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.     

On sparse estimation for semiparametric linear transformation models

Semiparametric linear transformation models have received much attention due to their high flexibility in modeling survival data. A useful estimating equation procedure was recently proposed by Chen et al. (2002) [21] for linear transformation models to jointly estimate parametric and nonparametric terms. They showed that this procedure can yield a consistent and robust estimator. However, the problem of variable selection for linear transformation models has been less studied, partially because a convenient loss function is not readily available under this context. In this paper, we propose a simple yet powerful approach to achieve both sparse and consistent estimation for linear transformation models. The main idea is to derive a profiled score from the estimating equation of Chen et al. [21], construct a loss function based on the profile scored and its variance, and then minimize the loss subject to some shrinkage penalty. Under regularity conditions, we have shown that the resulting estimator is consistent for both model estimation and variable selection. Furthermore, the estimated parametric terms are asymptotically normal and can achieve a higher efficiency than that yielded from the estimation equations. For computation, we suggest a one-step approximation algorithm which can take advantage of the LARS and build the entire solution path efficiently. Performance of the new procedure is illustrated through numerous simulations and real examples including one microarray data.     

Statistical inference on parametric part for partially linear single-index model

Statistical inference on parametric part for the partially linear single-index model (PLSIM) is considered in this paper. A profile least-squares technique for estimating the parametric part is proposed and the asymptotic normality of the profile least-squares estimator is given. Based on the estimator, a generalized likelihood ratio (GLR) test is proposed to test whether parameters on linear part for the model is under a contain linear restricted condition. Under the null model, the proposed GLR statistic follows asymptotically the χ²-distribution with the scale constant and degree of freedom independent of the nuisance parameters, known as Wilks phenomenon. Both simulated and real data examples are used to illustrate our proposed methods.     

部分线性模型的随机约束岭估计

研究了部分线性回归模型附加有随机约束条件时的估计问题.基于Profile最小二乘方法和混合估计方法提出了参数分量随机约束下的Profile混合估计,并研究了其性质.为了克服共线性问题,构造了参数分量的Profile混合岭估计,并给出了估计量的偏和方差.     

Jackknifing in partially linear regression models with serially correlated errors

In this paper jackknifing technique is examined for functions of the parametric component in a partially linear regression model with serially correlated errors. By deleting partial residuals a jackknife-type estimator is proposed. It is shown that the jackknife-type estimator and the usual semiparametric least-squares estimator (SLSE) are asymptotically equivalent. However, simulation shows that the former has smaller biases than the latter when the sample size is small or moderate. Moreover, since the errors are correlated, both the Tukey type and the delta type jackknife asymptotic variance estimators are not consistent. By introducing cross-product terms, a consistent estimator of the jackknife asymptotic variance is constructed and shown to be robust against heterogeneity of the error variances. In addition, simulation results show that confidence interval estimation based on the proposed jackknife estimator has better coverage probability than that based on the SLSE, even though the latter uses the information of the error structure, while the former does not.     

Bernstein Polynomial Estimation in the Partially Linear Model under Monotonicity Constraints

In this paper, a Bernstein-polynomial-based likelihood method is proposed for the partially linear model under monotonicity constraints. Monotone Bernstein polynomials are employed to approximate the monotone nonparametric function in the model. The estimator of the regression parameter is shown to be asymptotically normal and efficient, and the rate of convergence of the estimator of the nonparametric component is established, which could be the optimal under the smooth assumptions. A simulation study and a real data analysis are conducted to evaluate the finite sample performance of the proposed method.     

变系数部分线性模型的加权随机约束s-K估计

研究随机约束条件下半参数变系数部分线性模型的参数估计问题,当回归模型线性部分变量存在多重共线性时,基于Profile最小二乘方法、s-K估计和加权混合估计构造参数向量的加权随机约束s-K估计,并在均方误差矩阵准则下给出新估计量优于s-K估计和加权混合估计的充要条件,最后通过蒙特卡洛数值模拟验证所提出估计量的有限样本性质.     

Corrected-loss estimation for Error-in-Variable partially linear model

We consider an Error-in-Variable partially linear model where the covariates of linear part are measured with error which follows a normal distribution with a known covariance matrix. We propose a corrected-loss estimation of the covariate effect. The proposed estimator is asymptotically normal. Simulation studies are presented to show that the proposed method performs well with finite samples, and the proposed method is applied to a real data set.     

The Multivariate Partially Linear EV Model with B-spline

A multivariate partially linear EV model is considered in this paper. By correcting the attenuation, a modified B-spline least squares estimator for both the parametric and the nonparametric components is proposed. Moreover, we investigate the asymptotical normality of the modified estimator of the parametric components and the convergence rate of the estimator of the nonparametric function.     

Statistical inference in partially-varying-coefficient single-index model

Consider a varying-coefficient single-index model which consists of two parts: the linear part with varying coefficients and the nonlinear part with a single-index structure, and are hence termed as varying-coefficient single-index models. This model includes many important regression models such as single-index models, partially linear single-index models, varying-coefficient model and varying-coefficient partially linear models as special examples. In this paper, we mainly study estimating problems of the varying-coefficient vector, the nonparametric link function and the unknown parametric vector describing the single-index in the model. A stepwise approach is developed to obtain asymptotic normality estimators of the varying-coefficient vector and the parametric vector, and estimators of the nonparametric link function with a convergence rate. The consistent estimator of the structural error variance is also obtained. In addition, asymptotic pointwise confidence intervals and confidence regions are constructed for the varying coefficients and the parametric vector. The bandwidth selection problem is also considered. A simulation study is conducted to evaluate the proposed methods, and real data analysis is also used to illustrate our methods.     

Positive shrinkage, improved pretest and absolute penalty estimators in partially linear models

Shrinkage estimators of a partially linear regression parameter vector are constructed by shrinking estimators in the direction of the estimate which is appropriate when the regression parameters are restricted to a linear subspace. We investigate the asymptotic properties of positive Stein-type and improved pretest semiparametric estimators under quadratic loss. Under an asymptotic distributional quadratic risk criterion, their relative dominance picture is explored analytically. It is shown that positive Stein-type semiparametric estimators perform better than the usual Stein-type and least square semiparametric estimators and that an improved pretest semiparametric estimator is superior to the usual pretest semiparametric estimator. We also consider an absolute penalty type estimator for partially linear models and give a Monte Carlo simulation comparisons of positive shrinkage, improved pretest and the absolute penalty type estimators. The comparison shows that the shrinkage method performs better than the absolute penalty type estimation method when the dimension of the parameter space is much larger than that of the linear subspace.     

多元部分线性模型的B-样条估计(英文)

本文考虑多元部分线性回归模型的估计问题,得到了该模型参数的最小二乘估计和非参数函数的B-样条估计,并证明了参数估计的渐近正态性,给出了非参数函数估计的最优收敛速度.     

Efficiency of test for independence after Box-Cox transformation

We consider the efficiency and the power of the normal theory test for independence after a Box-Cox transformation. We obtain an expression for the correlation between the variates after a Box-Cox transformation in terms of the correlation on the normal scale. We discuss the efficiency of test of independence after a Box-Cox transformation and show that for the family considered it is always more efficient to conduct the test of independence based on Pearson correlation coefficient after transformation to normality. Power of test of independence before and after a Box-Cox transformation is studied for a finite sample size using Monte Carlo simulation. Our results show that we can increase the power of the normal-theory test for independence after estimating the transformation parameter from the data. The procedure has application for generating non-negative random variables with prescribed correlation.