Bounded-leverage weights for robust regression estimators

Both the least squares estimator and M-estimators of regression coefficients are susceptible to distortion when high leverage points occur among the predictor variables in a multiple linear regression model. In this article a weighting scheme which enables one to bound the leverage values of a weighted matrix of predictor variables is proposed. Bounded-leverage weighting of the predictor variables followed by M-estimation of the regression coefficients is shown to be effective in protecting against distortion due to extreme predictor-variable values, extreme response values, or outlier-induced multieollinearites. Bounded-leverage estimators can also protect against distortion by small groups of high leverage points.     

Linearly admissible estimators of stochastic regression coefficient under balanced loss function

The admissibility of linear estimators in a linear model with stochastic regression coefficient is investigated under a balanced loss function. The sufficient and necessary conditions for linear estimators to be admissible in classes of homogeneous and non-homogeneous linear estimators are obtained, respectively.     

A simulation study of five biased estimators for straight line regression

Five biased estimators of the slope in straight line regression are considered. For each, the estimate of the “bias parameter”, k, is a function of N, the number of observations, and [rcirc]² , the square of the least squares estimate of the standardized slope, β. The estimators include that of Farebrother, the ridge estimator of Hoerl, Kennard, and Baldwin, Vinod's shrunken estimators., and a new modification of one of the latter. Properties of the estimators are studied for 13 combinations of N and 3. Results of simulation experiments provide empirical evidence concerning the values of means and variances of the biased estimators of the slope and estimates of the “bias parameter”, the mean square errors of the estimators, and the frequency of improvement relative to least squares. Adjustments to degrees of freedom in the biased regression analysis of variance table are also considered. An extension of the new modification to the case of p> 1 independent variables is presented in an Appendix.     

A simulation study of ridge and other regression estimators

We consider a number of estimators of regression coefficients, all of generalized ridge, or 'shrinkage' type. Results of a simulation study indicate that with respect to two commonly used mean square error criteria, two ordinary ridge estimators, one proposed by Hoerl, Kennard and Baldwin, and the other introduced here, perform substantially better than both least squares and the other estimators discussed here     

A comparison of biased regression estimators using a pitman nearness criterion

Biased regression estimators have traditionally benn studied using the Mean Square Error (MSE) criterion. Usually these comparisons have been based on the sum of the MSE's of each of the individual parameters, i.e., a scaler valued measure that is the trace of the MSE matrix. However, since this summed MSE does not consider the covariance structure of the estimators, we propose the use of a Pitman Measure of Closeness (PMC) criterion (Keating and Gupta, 1984; Keating and Mason, 1985). In this paper we consider two versions of PMC. One of these compares the estimates and the other compares the resultant predicted values for 12 different regression estimators. These estimators represent three classes of estimators, namely, ridge, shrunken, and principal component estimators. The comparisons of these estimators using the PMC criteria are contrasted with the usual MSE criteria as well as the prediction mean square error. Included in the estimators is a relatively new estimator termed the generalized principal component estimator proposed by Jolliffe. This estimator has previously received little attention in the literature.     

Variable selection for semiparametric varying coefficient partially linear model based on modal regression with missing data

Abstract

In this article, we focus on the variable selection for semiparametric varying coefficient partially linear model with response missing at random. Variable selection is proposed based on modal regression, where the non parametric functions are approximated by B-spline basis. The proposed procedure uses SCAD penalty to realize variable selection of parametric and nonparametric components simultaneously. Furthermore, we establish the consistency, the sparse property and asymptotic normality of the resulting estimators. The penalty estimation parameters value of the proposed method is calculated by EM algorithm. Simulation studies are carried out to assess the finite sample performance of the proposed variable selection procedure.     

Linearly admissible estimators on linear functions of regression coefficient under balanced loss function

Linearly admissible estimators on linear functions of regression coefficient are studied in a singular linear model and balanced loss when the design matrix has not full column rank. The sufficient and necessary conditions for linear estimators to be admissible are obtained respectively in homogeneous and inhomogeneous classes.     

Moment properties of estimators for a type 1 extreme-value regression model

A regression model is considered in which the response variable has a type 1 extreme-value distribution for smallest values. Bias approximations for the maximum likelihood estimators are pivm and a bias reduction estimator for the scale parameter is proposed. The small sample moment properties of the maximum likelihood estimators are compared with the properties of the ordinary least squares estimators and the best linear unbiased estimators based on order statistics for grouped data.     

Convergence rates for uniform confidence intervals based on local polynomial regression estimators

We investigate the convergence rates of uniform bias-corrected confidence intervals for a smooth curve using local polynomial regression for both the interior and boundary region. We discuss the cases when the degree of the polynomial is odd and even. The uniform confidence intervals are based on the volume-of-tube formula modified for biased estimators. We empirically show that the proposed uniform confidence intervals attain, at least approximately, nominal coverage. Finally, we investigate the performance of the volume-of-tube based confidence intervals for independent non-Gaussian errors.     

A comparison of estimators of a common correlation coefficient

It is often of interest to test the hypothesis that all off-diagonal elements of the correlation matrix of a multivariate normal distribution are equal. If the hypothesis of equal correlation can be accepted, it then may be of interest to estimate the common correlation coefficient. In this paper, four estimators of the common correlation are compared in terms of bias, variance, mean squared error, adequacy of the normal approximation, and ease of calculation. The average sample correlation is seen to be comparable to the other estimators and is recommended here since it is the easiest to calculate. The estimators are compared using simulation.     

A Jackknifed estimators for the negative binomial regression model

Shrinkage estimator is a commonly applied solution to the general problem caused by multicollinearity. Recently, the ridge regression (RR) estimators for estimating the ridge parameter k in the negative binomial (NB) regression have been proposed. The Jackknifed estimators are obtained to remedy the multicollinearity and reduce the bias. A simulation study is provided to evaluate the performance of estimators. Both mean squared error (MSE) and the percentage relative error (PRE) are considered as the performance criteria. The simulated result indicated that some of proposed Jackknifed estimators should be preferred to the ML method and ridge estimators to reduce MSE and bias.     

Parametric simultaneous robust inferences for regression coefficient under generalized linear models

In this article, the parametric robust regression approaches are proposed for making inferences about regression parameters in the setting of generalized linear models (GLMs). The proposed methods are able to test hypotheses on the regression coefficients in the misspecified GLMs. More specifically, it is demonstrated that with large samples, the normal and gamma regression models can be properly adjusted to become asymptotically valid for inferences about regression parameters under model misspecification. These adjusted regression models can provide the correct type I and II error probabilities and the correct coverage probability for continuous data, as long as the true underlying distributions have finite second moments.     

Empirical likelihood for linear regression models under imputation for missing responses

The authors study the empirical likelihood method for linear regression models. They show that when missing responses are imputed using least squares predictors, the empirical log‐likelihood ratio is asymptotically a weighted sum of chi‐square variables with unknown weights. They obtain an adjusted empirical log‐likelihood ratio which is asymptotically standard chi‐square and hence can be used to construct confidence regions. They also obtain a bootstrap empirical log‐likelihood ratio and use its distribution to approximate that of the empirical log‐likelihood ratio. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths of confidence intervals, and perform better than a normal approximation based method.     

Bias of maximum likelihood estimators of the response standard deviation in regression

The bias of maximum likelihood estimators of the standard deviation of the response in location/scale regression models is considered. Results are obtained for a very wide family of densities for the response variable. These are used to propose point estimators with improved mean square error properties and to demonstrate the importance of bias correction in statistical inference when samples are moderately small.     

A Monte Carlo simulation study on partially adaptive estimators of linear regression models

This paper presents a comprehensive comparison of well-known partially adaptive estimators (PAEs) in terms of efficiency in estimating regression parameters. The aim is to identify the best estimators of regression parameters when error terms follow from normal, Laplace, Student's t, normal mixture, lognormal and gamma distribution via the Monte Carlo simulation. In the results of the simulation, efficient PAEs are determined in the case of symmetric leptokurtic and skewed leptokurtic regression error data. Additionally, these estimators are also compared in terms of regression applications. Regarding these applications, using certain standard error estimators, it is shown that PAEs can reduce the standard error of the slope parameter estimate relative to ordinary least squares.     

The estimation of missing observations in related time series data: Further results

Earlier work has provided an efficient method for the prediction of missing data in a dependent variable series using a system of grouped regression equations. This paper extends the previous literature in two ways. First, a test statistic capable of indicating the advantage of the grouped procedure is derived. Second, it is demonstrated through an empirical application that the most prevalent methodology used for examining the impact of financial economic events is a special case of the missing data estimation problem.     

Improving the estimators of the parameters of a probit regression model: A ridge regression approach

This paper considered the estimation of the regression parameters of a general probit regression model. Accordingly, we proposed five ridge regression (RR) estimators for the probit regression models for estimating the parameters (β)$(\beta )$ when the weighted design matrix is ill-conditioned and it is suspected that the parameter β$\beta$ may belong to a linear subspace defined by Hβ=h$H\beta =h$. Asymptotic properties of the estimators are studied with respect to quadratic biases, MSE matrices and quadratic risks. The regions of optimality of the proposed estimators are determined based on the quadratic risks. Some relative efficiency tables and risk graphs are provided to illustrate the numerical comparison of the estimators. We conclude that when q≥3$q\ge 3$, one would uses PRRRE; otherwise one uses PTRRE with some optimum size α$\alpha$. We also discuss the performance of the proposed estimators compare to the alternative ridge regression method due to Liu (1993).     

Choosing shrinkage estimators for regression problems

A Bayesian formulation of the canonical form of the standard regression model is used to compare various Stein-type estimators and the ridge estimator of regression coefficients, A particular (“constant prior”) Stein-type estimator having the same pattern of shrinkage as the ridge estimator is recommended for use.     

A comparison of estimators for regression models with change points

We consider two problems concerning locating change points in a linear regression model. One involves jump discontinuities (change-point) in a regression model and the other involves regression lines connected at unknown points. We compare four methods for estimating single or multiple change points in a regression model, when both the error variance and regression coefficients change simultaneously at the unknown point(s): Bayesian, Julious, grid search, and the segmented methods. The proposed methods are evaluated via a simulation study and compared via some standard measures of estimation bias and precision. Finally, the methods are illustrated and compared using three real data sets. The simulation and empirical results overall favor both the segmented and Bayesian methods of estimation, which simultaneously estimate the change point and the other model parameters, though only the Bayesian method is able to handle both continuous and dis-continuous change point problems successfully. If it is known that regression lines are continuous then the segmented method ranked first among methods.