Generalized ridge regression estimators under the LINEX loss function

In this paper, we examine the risk performance of the generalized ridge regression (GRR) and feasible GRR estimators when the LINEX loss function is used. A sufficient condition for the GRR estimator to dominate the OLS estimator is shown, and the risk functions of the feasible GRR estimator and the OLS estimator are numerically compared.     

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.     

On the almost unbiased ridge regression estimator

The purpose of this paper is two-fold. One is to compare the almost unbiased generalized ridge regression (AUGRR) estimator proposed by Singh, Chaubey and Dwivedi (1986) with the generalized ridge regression (GRR) estimator and with the ordinary least squares (OLS) estimator in terms of the mean squared error criterion. Second is to examine small sample properties of the operational almost unbiased ordinary ridge regression (AUORR) estimator by Monte Carlo experiments.     

Ridge regression estimators in the linear regression models with non-spherical errors

The present paper considers a family of ordinary ridge regression estimators in the linear regression model when the disturbances covariance matrix depends upon a few unknown parameters. An asymptotic expansion for the distribution of the ridge regression estimator is developed and under the quadratic loss function its asymptotic risk is compared with that of the feasible GLS estimator.     

Preliminary test and Bayes Estimation of a Location Parameter Under Blinex Loss

In this article, the preliminary test estimator is considered under the BLINEX loss function. The problem under consideration is the estimation of the location parameter from a normal distribution. The risk under the null hypothesis for the preliminary test estimator, the exact risk function for restricted maximum likelihood and approximated risk function for the unrestricted maximum likelihood estimator, are derived under BLINEX loss and the different risk structures are compared to one another both analytically and computationally. As a motivation on the use of BLINEX rather than LINEX, the risk for the preliminary test estimator under BLINEX loss is compared to the risk of the preliminary test estimator under LINEX loss and it is shown that the LINEX expected loss is higher than BLINEX expected loss. Furthermore, two feasible Bayes estimators are derived under BLINEX loss, and a feasible Bayes preliminary test estimator is defined and compared to the classical preliminary test estimator.     

New biased estimators under the LINEX loss function

In this paper, using the asymmetric LINEX loss function we derive the risk function of the generalized Liu estimator and almost unbiased generalized Liu estimator. We also examine the risk performance of the feasible generalized Liu estimator and feasible almost unbiased generalized Liu estimator when the LINEX loss function is used.     

Estimating the density of a possibly missing response variable in nonlinear regression

This paper considers linear and nonlinear regression with a response variable that is allowed to be “missing at random”. The only structural assumptions on the distribution of the variables are that the errors have mean zero and are independent of the covariates. The independence assumption is important. It enables us to construct an estimator for the response density that uses all the observed data, in contrast to the usual local smoothing techniques, and which therefore permits a faster rate of convergence. The idea is to write the response density as a convolution integral which can be estimated by an empirical version, with a weighted residual-based kernel estimator plugged in for the error density. For an appropriate class of regression functions, and a suitably chosen bandwidth, this estimator is consistent and converges with the optimal parametric rate n^1/2. Moreover, the estimator is proved to be efficient (in the sense of Hájek and Le Cam) if an efficient estimator is used for the regression parameter.     

The error components regression model: conditional relative efficiency comparisons

This paper considers a simple linear regression with two-way error component disturbances and derives the conditional relative efficiency ofany feasible GLS estimator with respect to OLS, true GLS, orany other feasible GLS estimator, conditional on the estimated variance components. This is done at two crucial choices of the x variable. The first choice is where OLS is least efficient with respect to GLS and the second choice is where an arbitrary feasible GLS estimator is least efficient with respect to GLS. Our findings indicate that a better guess of a certain ‘variance components ratio’ leads to better estimates of the regression coefficients.     

Feasible robust estimator in restricted semiparametric regression models based on the LTS approach

Under some nonstochastic linear restrictions based on either additional information or prior knowledge in a semiparametric regression model, a family of feasible generalized robust estimators for the regression parameter is proposed. The least trimmed squares (LTS) method proposed by Rousseeuw as a highly robust regression estimator is a statistical technique for fitting a regression model based on the subset of h observations (out of n) whose least-square fit possesses the smallest sum of squared residuals. The coverage h may be set between n/2 and n. The LTS estimator involves computing the hyperplane that minimizes the sum of the smallest h squared residuals. For practical purpose, it is assumed that the covariance matrix of the error term is unknown and thus feasible estimators are replaced. Then, we develop an algorithm for the LTS estimator based on feasible methods. Through the Monte Carlo simulation studies and a real data example, performance of the feasible type of robust estimators is compared with the classical ones in restricted semiparametric regression models.     

The feasible generalized restricted ridge regression estimator

The presence of autocorrelation in errors and multicollinearity among the regressors have undesirable effects on the least-squares regression. There are a wide range of methods which are proposed to overcome the usefulness of the ordinary least-squares estimator or the generalized least-squares estimator, such as the Stein-rule, restricted least-squares or ridge estimator. Therefore, we introduce a new feasible generalized restricted ridge regression (FGRR) estimator to examine multicollinearity and autocorrelation problems simultaneously for the general linear regression model. We also derive some statistical properties of the FGRR estimator and comparisons have been conducted using matrix mean-square error. Moreover, a Monte Carlo simulation experiment is performed to investigate the performance of the proposed estimator over the others.     

Rank-based ridge estimation in multiple linear regression

Multicollinearity and model misspecification are frequently encountered problems in practice that produce undesirable effects on classical ordinary least squares (OLS) regression estimator. The ridge regression estimator is an important tool to reduce the effects of multicollinearity, but it is still sensitive to a model misspecification of error distribution. Although rank-based statistical inference has desirable robustness properties compared to the OLS procedures, it can be unstable in the presence of multicollinearity. This paper introduces a rank regression estimator for regression parameters and develops tests for general linear hypotheses in a multiple linear regression model. The proposed estimator and the tests have desirable robustness features against the multicollinearity and model misspecification of error distribution. Asymptotic behaviours of the proposed estimator and the test statistics are investigated. Real and simulated data sets are used to demonstrate the feasibility and the performance of the estimator and the tests.     

An omnibus test of goodness-of-fit for conditional distributions with applications to regression models

We introduce an omnibus goodness-of-fit test for statistical models for the conditional distribution of a random variable. In particular, this test is useful for assessing whether a regression model fits a data set on all its assumptions. The test is based on a generalization of the Cramér–von Mises statistic and involves a local polynomial estimator of the conditional distribution function. First, the uniform almost sure consistency of this estimator is established. Then, the asymptotic distribution of the test statistic is derived under the null hypothesis and under contiguous alternatives. The extension to the case where unknown parameters appear in the model is developed. A simulation study shows that the test has good power against some common departures encountered in regression models. Moreover, its power is comparable to that of other nonparametric tests designed to examine only specific departures.     

A study on the least squares estimator of multivariate isotonic regression function

Consider the problem of pointwise estimation of f in a multivariate isotonic regression model Z=f(X₁,…,X_d)+ϵ, where Z is the response variable, f is an unknown nonparametric regression function, which is isotonic with respect to each component, and ϵ is the error term. In this article, we investigate the behavior of the least squares estimator of f. We generalize the greatest convex minorant characterization of isotonic regression estimator for the multivariate case and use it to establish the asymptotic distribution of properly normalized version of the estimator. Moreover, we test whether the multivariate isotonic regression function at a fixed point is larger (or smaller) than a specified value or not based on this estimator, and the consistency of the test is established. The practicability of the estimator and the test are shown on simulated and real data as well.     

Nonstationary nonlinear quantile regression

This study examines estimation and inference based on quantile regression for parametric nonlinear models with an integrated time series covariate. We first derive the limiting distribution of the nonlinear quantile regression estimator and then consider testing for parameter restrictions, when the regression function is specified as an asymptotically homogeneous function. We also study linear-in-parameter regression models when the regression function is given by integrable regression functions as well as asymptotically homogeneous regression functions. We, furthermore, propose a fully modified estimator to reduce the bias in the original estimator under a certain set of conditions. Finally, simulation studies show that the estimators behave well, especially when the regression error term has a fat-tailed distribution.     

A robust and efficient estimation method for nonparametric models with jump points

Nonparametric models with jump points have been considered by many researchers. However, most existing methods based on least squares or likelihood are sensitive when there are outliers or the error distribution is heavy tailed. In this article, a local piecewise-modal method is proposed to estimate the regression function with jump points in nonparametric models, and a piecewise-modal EM algorithm is introduced to estimate the proposed estimator. Under some regular conditions, the large-sample theory is established for the proposed estimators. Several simulations are presented to evaluate the performances of the proposed method, which shows that the proposed estimator is more efficient than the local piecewise-polynomial regression estimator in the presence of outliers or heavy tail error distribution. What is more, the proposed procedure is asymptotically equivalent to the local piecewise-polynomial regression estimator under the assumption that the error distribution is a Gaussian distribution. The proposed method is further illustrated via the sea-level pressures.     

A Wald-type test statistic based on robust modified median estimator in logistic regression models

In this paper a new robust estimator, modified median estimator, is introduced and studied for the logistic regression model. This estimator is based on the median estimator considered in Hobza et al. [Robust median estimator in logistic regression. J Stat Plan Inference. 2008;138:3822–3840]. Its asymptotic distribution is obtained. Using the modified median estimator, we also consider a Wald-type test statistic for testing linear hypotheses in the logistic regression model and we obtain its asymptotic distribution under the assumption of random regressors. An extensive simulation study is presented in order to analyse the efficiency as well as the robustness of the modified median estimator and Wald-type test based on it.     

On risk comparisons of some estimators in a linear regression model under inagaki’S loss function and nonnormal error terms

In this paper we consider the risk performances of some estimators for both location and scale parameters in a linear regression model under Inagaki’s loss function We prove that the pre-test estimator for location parameter is dominated by the Stein-rule estimator under Inagaki’s loss function when the distribution of error terms is expressed by the scale mixture of normal distribution and the variance of error terms is unknown.. It is an extension of the results in Nagata (1983) to our situation Also we perform numerical calculations to draw the shapes of the risks.     

Factor analysis regression

In the presence of multicollinearity the literature points to principal component regression (PCR) as an estimation method for the regression coefficients of a multiple regression model. Due to ambiguities in the interpretation, involved by the orthogonal transformation of the set of explanatory variables, the method could not yet gain wide acceptance. Factor analysis regression (FAR) provides a model-based estimation method which is particularly tailored to overcome multicollinearity in an errors-in-variables setting. In this paper two feasible versions of a FAR estimator are compared with the OLS estimator and the PCR estimator by means of Monte Carlo simulation. While the PCR estimator performs best in cases of strong and high multicollinearity, the Thomson-based FAR estimator proves to be superior when the regressors are moderately correlated.     

Robust Estimation of Multivariate Linear Model Based on Depth Weighted Mean and Scatter

Based on the projection depth weighted mean and scatter estimation of the joint distribution of (x, y), we introduce a robust estimator of the regression coefficients for the multivariate linear model. The new estimator possesses desirable properties including affine invariance, Fisher consistency, and asymptotic normality. Also, we study the robustness of the estimator in terms of breakdown point and influence function. Extensive simulation studies are performed to investigate the finite sample behavior of robustness and efficiency. The methodology is illustrated with a real data example.     

The exact distribution and density functions of the stein-type estimator for normal variance

In this paper, we derive the exact distribution and density functions of the Stein-type estimator for the normal variance. It is shown by numerical evaluation that the density function of the Stein-type estimator is unimodal and concentrates around the mode more than that of the usual estimator.