Linearized Ridge Regression Estimator in Linear Regression

In this article, we aim to study the linearized ridge regression (LRR) estimator in a linear regression model motivated by the work of Liu (1993). The LRR estimator and the two types of generalized Liu estimators are investigated under the PRESS criterion. The method of obtaining the optimal generalized ridge regression (GRR) estimator is derived from the optimal LRR estimator. We apply the Hald data as a numerical example and then make a simulation study to show the main results. It is concluded that the idea of transforming the GRR estimator as a complicated function of the biasing parameters to a linearized version should be paid more attention in the future.     

A Generalized Stochastic Restricted Ridge Regression Estimator

In this article, we introduce a new stochastic restricted estimator for the unknown vector parameter in the linear regression model when stochastic linear restrictions on the parameters hold. We show that the new estimator is a generalization of the ordinary mixed estimator (OME), Liu estimator (LE), ordinary ridge estimator (ORR), (k-d) class estimator, stochastic restricted Liu estimator (SRLE), and stochastic restricted ridge estimator (SRRE). Performance of the new estimator in comparison to other estimators in terms of the mean squares error matrix (MMSE) is examined. Numerical example from literature have been given to illustrate the results.     

Linearized Restricted Ridge Regression Estimator in Linear Regression

This article primarily aims to put forward the linearized restricted ridge regression (LRRR) estimator in linear regression models. Two types of LRRR estimators are investigated under the PRESS criterion and the optimal LRRR estimators and the optimal restricted generalized ridge regression estimator are obtained. We apply the results to the Hald data and finally make a simulation study by using the method of McDonald and Galarneau.     

Non-Diagonal-Type Estimator in Linear Regression

This short article mainly aims to introduce the notion of the non-diagonal-type estimator (NDTE) by means of the singular value decomposition theorem in the linear regression model to improve some classical linear estimators that can be called the diagonal-type estimators. We derive the optimal NDTE under the mean squared error criterion and its iterative version through matrix techniques. A simulation study is finally conducted to illustrate the theoretical results.     

Robust Linearized Ridge M-estimator for Linear Regression Model

In the multiple linear regression, multicollinearity and outliers are commonly occurring problems. They produce undesirable effects on the ordinary least squares estimator. Many alternative parameter estimation methods are available in the literature which deals with these problems independently. In practice, it may happen that the multicollinearity and outliers occur simultaneously. In this article, we present a new estimator called as Linearized Ridge M-estimator which combats the problem of simultaneous occurrence of multicollinearity and outliers. A real data example and a simulation study is carried out to illustrate the performance of the proposed estimator.     

Mixed Discrete and Continuous Cox Regression Model

The Cox (1972) regression model is extended to include discrete and mixed continuous/discrete failure time data by retaining the multiplicative hazard rate form of the absolutely continuous model. Application of martingale arguments to the regression parameter estimating function show the Breslow (1974) estimator to be consistent and asymptotically Gaussian under this model. A computationally convenient estimator of the variance of the score function can be developed, again using martingale arguments. This estimator reduces to the usual hypergeometric form in the special case of testing equality of several survival curves, and it leads more generally to a convenient consistent variance estimator for the regression parameter. A small simulation study is carried out to study the regression parameter estimator and its variance estimator under the discrete Cox model special case and an application to a bladder cancer recurrence dataset is provided.     

Multiplicative Adjustment Method for Semiparametric Regression with Mixing Dependent Data

In this article, we extend a semiparametric regression estimator with multiplicative adjustment to time series context. The asymptotic theory and results from a simulation study are discussed. Theoretical results and numerical comparison show that, in the time series case, the semiparametric estimator is better than the traditional local polynomial estimator in a wide neighbourhood around the true regression function.     

A Generalized Diagonal Ridge-type Estimator in Linear Regression

This article introduces a general class of biased estimator, namely a generalized diagonal ridge-type (GDR) estimator, for the linear regression model when multicollinearity occurs. The estimator represents different kinds of biased estimators when different parameters are obtained. Some properties of this estimator are discussed and an iterative procedure is provided for selecting the parameters. A Monte Carlo simulation study and an application show that the GDR estimator performs much better than the ordinary least squares (OLS) estimator under the mean square error (MSE) criterion when severe multicollinearity is present.     

Guided Censored Regression

Parametrically guided non‐parametric regression is an appealing method that can reduce the bias of a non‐parametric regression function estimator without increasing the variance. In this paper, we adapt this method to the censored data case using an unbiased transformation of the data and a local linear fit. The asymptotic properties of the proposed estimator are established, and its performance is evaluated via finite sample simulations.     

Regression with outlier shrinkage

We propose a robust regression method called regression with outlier shrinkage (ROS) for the traditional n>p$n>p$ cases. It improves over the other robust regression methods such as least trimmed squares (LTS) in the sense that it can achieve maximum breakdown value and full asymptotic efficiency simultaneously. Moreover, its computational complexity is no more than that of LTS. We also propose a sparse estimator, called sparse regression with outlier shrinkage (SROS), for robust variable selection and estimation. It is proven that SROS can not only give consistent selection but also estimate the nonzero coefficients with full asymptotic efficiency under the normal model. In addition, we introduce a concept of nearly regression equivariant estimator for understanding the breakdown properties of sparse estimators, and prove that SROS achieves the maximum breakdown value of nearly regression equivariant estimators. Numerical examples are presented to illustrate our methods.     

多辅助变量线性组合的回归估计

以提高估计量的精度为目的,定义了一种新的多个辅助变量的回归估计法,从理论上研究了该方法下权的选取方法,并将该方法下的估计量与Raj多辅助变量回归估计量、Ghosh多元线性回归估计量在精度上进行了数值比较,结果表明：这种新的多辅助变量的回归估计法在精度上优于Raj多辅助变量回归估计和Ghosh多元线性回归估计。     

Nonlinear Quantile Regression Estimation of Longitudinal Data

This article examines a weighted version of the quantile regression estimator as defined by Koenker and Bassett (1978 Koenker , R. , Bassett , G. ( 1978 ). Regression quantiles . Econometrica 46 : 33 – 50 .[Crossref], [Web of Science ®] , [Google Scholar]), adjusted to the case of nonlinear longitudinal data. Using a four-parameter logistic growth function and error terms following an AR(1) model, different weights are used and compared in a simulation study. The findings indicate that the nonlinear quantile regression estimator is performing well, especially for the median regression case, that the differences between the weights are small, and that the estimator performs better when the correlation in the AR(1) model increases. A comparison is also made with the corresponding mean regression estimator, which is found to be less robust. Finally, the estimator is applied to a data set with growth patterns of two genotypes of soybean, which gives some insights into how the quantile regressions provide a more complete picture of the data than the mean regression.     

Estimation and Test for Multi-Dimensional Regression Models

This work is concerned with the estimation of multi-dimensional regression and the asymptotic behavior of the test involved in selecting models. The main problem with such models is that we need to know the covariance matrix of the noise to get an optimal estimator. We show in this article that if we choose to minimize the logarithm of the determinant of the empirical error covariance matrix, then we get an asymptotically optimal estimator. Moreover, under suitable assumptions, we show that this cost function leads to a very simple asymptotic law for testing the number of parameters of an identifiable and regular regression model. Numerical experiments confirm the theoretical results.     

Two Stochastic Restricted Principal Components Regression Estimator in Linear Regression

In this article, we propose two stochastic restricted principal components regression estimator by combining the approach followed in obtaining the ordinary mixed estimator and the principal components regression estimator in linear regression model. The performance of the two new estimators in terms of matrix MSE criterion is studied. We also give an example and a Monte Carlo simulation to show the theoretical results.     

A Tobit Ridge Regression Estimator

This article analyzes the effects of multicollienarity on the maximum likelihood (ML) estimator for the Tobit regression model. Furthermore, a ridge regression (RR) estimator is proposed since the mean squared error (MSE) of ML becomes inflated when the regressors are collinear. To investigate the performance of the traditional ML and the RR approaches we use Monte Carlo simulations where the MSE is used as performance criteria. The simulated results indicate that the RR approach should always be preferred to the ML estimation method.     

New Ridge Regression Estimator in Semiparametric Regression Models

In the context of ridge regression, the estimation of shrinkage parameter plays an important role in analyzing data. Many efforts have been put to develop the computation of risk function in different full-parametric ridge regression approaches using eigenvalues and then bringing an efficient estimator of shrinkage parameter based on them. In this respect, the estimation of shrinkage parameter is neglected for semiparametric regression model. Not restricted, but the main focus of this approach is to develop necessary tools for computing the risk function of regression coefficient based on the eigenvalues of design matrix in semiparametric regression. For this purpose the differencing methodology is applied. We also propose a new estimator for shrinkage parameter which is of harmonic type mean of ridge estimators. It is shown that this estimator performs better than all the existing ones for the regression coefficient. For our proposal, a Monte Carlo simulation study and a real dataset analysis related to housing attributes are conducted to illustrate the efficiency of shrinkage estimators based on the minimum risk and mean squared error criteria.     

A New Regression Model: Modal Linear Regression

The mode of a distribution provides an important summary of data and is often estimated on the basis of some non‐parametric kernel density estimator. This article develops a new data analysis tool called modal linear regression in order to explore high‐dimensional data. Modal linear regression models the conditional mode of a response Y given a set of predictors x as a linear function of x . Modal linear regression differs from standard linear regression in that standard linear regression models the conditional mean (as opposed to mode) of Y as a linear function of x . We propose an expectation–maximization algorithm in order to estimate the regression coefficients of modal linear regression. We also provide asymptotic properties for the proposed estimator without the symmetric assumption of the error density. Our empirical studies with simulated data and real data demonstrate that the proposed modal regression gives shorter predictive intervals than mean linear regression, median linear regression and MM‐estimators.     

Nonparametric Wavelet Regression Based on Biased Data

The estimation of the regression function in the biased nonparametric regression model is investigated. We propose and develop a new wavelet-based methodology for this problem. In particular, an adaptive hard thresholding wavelet estimator is constructed. Under mild assumptions on the model, we prove that it enjoys powerful mean integrated squared error properties over Besov balls.     

Confidence Corridors for Multivariate Generalized Quantile Regression

We focus on the construction of confidence corridors for multivariate nonparametric generalized quantile regression functions. This construction is based on asymptotic results for the maximal deviation between a suitable nonparametric estimator and the true function of interest, which follow after a series of approximation steps including a Bahadur representation, a new strong approximation theorem, and exponential tail inequalities for Gaussian random fields. As a byproduct we also obtain multivariate confidence corridors for the regression function in the classical mean regression. To deal with the problem of slowly decreasing error in coverage probability of the asymptotic confidence corridors, which results in meager coverage for small sample sizes, a simple bootstrap procedure is designed based on the leading term of the Bahadur representation. The finite-sample properties of both procedures are investigated by means of a simulation study and it is demonstrated that the bootstrap procedure considerably outperforms the asymptotic bands in terms of coverage accuracy. Finally, the bootstrap confidence corridors are used to study the efficacy of the National Supported Work Demonstration, which is a randomized employment enhancement program launched in the 1970s. This article has supplementary materials online.     

空间回归模型估计中的最小二乘法

空间回归模型由于引入了空间地理信息而使得其参数估计变得复杂,因为主要采用最大似然法,致使一般人认为在空间回归模型参数估计中不存在最小二乘法。通过分析空间回归模型的参数估计技术,研究发现,最小二乘法和最大似然法分别用于估计空间回归模型的不同的参数,只有将两者结合起来才能快速有效地完成全部的参数估计。数理论证结果表明,空间回归模型参数最小二乘估计量是最佳线性无偏估计量。空间回归模型的回归参数可以在估计量为正态性的条件下而实施显著性检验,而空间效应参数则不可以用此方法进行检验。