A comparison of some robust,adaptive, and partially adaptive estimators of regression models

Numerous estimation techniques for regression models have been proposed. These procedures differ in how sample information is used in the estimation procedure. The efficiency of least squares (OLS) estimators implicity assumes normally distributed residuals and is very sensitive to departures from normality, particularly to "outliers" and thick-tailed distributions. Lead absolute deviation (LAD) estimators are less sensitive to outliers and are optimal for laplace random disturbances, but not for normal errors. This paper reports monte carlo comparisons of OLS,LAD, two robust estimators discussed by huber, three partially adaptiveestimators, newey's generalized method of moments estimator, and an adaptive maximum likelihood estimator based on a normal kernal studied by manski. This paper is the first to compare the relative performance of some adaptive robust estimators (partially adaptive and adaptive procedures) with some common nonadaptive robust estimators. The partially adaptive estimators are based on three flxible parametric distributions for the errors. These include the power exponential (Box-Tiao) and generalized t distributions, as well as a distribution for the errors, which is not necessarily symmetric. The adaptive procedures are "fully iterative" rather than one step estimators. The adaptive estimators have desirable large sample properties, but these properties do not necessarily carry over to the small sample case.

The monte carlo comparisons of the alternative estimators are based on four different specifications for the error distribution: a normal, a mixture of normals (or variance-contaminated normal), a bimodal mixture of normals, and a lognormal. Five hundred samples of 50 are used. The adaptive and partially adaptive estimators perform very well relative to the other estimation procedures considered, and preliminary results suggest that in some important cases they can perform much better than OLS with 50 to 80% reductions in standard errors.

     

Performance of Kibria's Method for the Heteroscedastic Ridge Regression Model: Some Monte Carlo Evidence

It is common for a linear regression model that the error terms display some form of heteroscedasticity and at the same time, the regressors are also linearly correlated. Both of these problems have serious impact on the ordinary least squares (OLS) estimates. In the presence of heteroscedasticity, the OLS estimator becomes inefficient and the similar adverse impact can also be found on the ridge regression estimator that is alternatively used to cope with the problem of multicollinearity. In the available literature, the adaptive estimator has been established to be more efficient than the OLS estimator when there is heteroscedasticity of unknown form. The present article proposes the similar adaptation for the ridge regression setting with an attempt to have more efficient estimator. Our numerical results, based on the Monte Carlo simulations, provide very attractive performance of the proposed estimator in terms of efficiency. Three different existing methods have been used for the selection of biasing parameter. Moreover, three different distributions of the error term have been studied to evaluate the proposed estimator and these are normal, Student's t and F distribution.     

A comparison of some robust, adaptive, and partially adaptive estimators of regression models

Numerous estimation techniques for regression models have been proposed. These procedures differ in how sample information is used in the estimation procedure. The efficiency of least squares (OLS) estimators implicity assumes normally distributed residuals and is very sensitive to departures from normality, particularly to "outliers" and thick-tailed distributions. Lead absolute deviation (LAD) estimators are less sensitive to outliers and are optimal for laplace random disturbances, but not for normal errors. This paper reports monte carlo comparisons of OLS,LAD, two robust estimators discussed by huber, three partially adaptiveestimators, newey's generalized method of moments estimator, and an adaptive maximum likelihood estimator based on a normal kernal studied by manski. This paper is the first to compare the relative performance of some adaptive robust estimators (partially adaptive and adaptive procedures) with some common nonadaptive robust estimators. The partially adaptive estimators are based on three flxible parametric distributions for the errors. These include the power exponential (Box-Tiao) and generalized t distributions, as well as a distribution for the errors, which is not necessarily symmetric. The adaptive procedures are "fully iterative" rather than one step estimators. The adaptive estimators have desirable large sample properties, but these properties do not necessarily carry over to the small sample case.

The monte carlo comparisons of the alternative estimators are based on four different specifications for the error distribution: a normal, a mixture of normals (or variance-contaminated normal), a bimodal mixture of normals, and a lognormal. Five hundred samples of 50 are used. The adaptive and partially adaptive estimators perform very well relative to the other estimation procedures considered, and preliminary results suggest that in some important cases they can perform much better than OLS with 50 to 80% reductions in standard errors.     

A new biased estimator based on ridge estimation

In this paper we introduce a new biased estimator for the vector of parameters in a linear regression model and discuss its properties. We show that our new biased estimator is superior, in the mean square error(mse) sense, to the ordinary least squares (OLS) estimator, the ordinary ridge regression (ORR) estimator and the Liu estimator. We also compare the performance of our new biased estimator with two other special Liu-type estimators proposed in Liu (2003). We illustrate our findings with a numerical example based on the widely analysed dataset on Portland cement.     

Comparisons of the <Emphasis Type="Italic">r</Emphasis> − <Emphasis Type="Italic">k</Emphasis> class estimator to the ordinary least squares estimator under the Pitman’s closeness criterion

In the presence of multicollinearity, the r − k class estimator is proposed as an alternative to the ordinary least squares (OLS) estimator which is a general estimator including the ordinary ridge regression (ORR), the principal components regression (PCR) and the OLS estimators. Comparison of competing estimators of a parameter in the sense of mean square error (MSE) criterion is of central interest. An alternative criterion to the MSE criterion is the Pitman’s (1937) closeness (PC) criterion. In this paper, we compare the r − k class estimator to the OLS estimator in terms of PC criterion so that we can get the comparison of the ORR estimator to the OLS estimator under the PC criterion which was done by Mason et al. (1990) and also the comparison of the PCR estimator to the OLS estimator by means of the PC criterion which was done by Lin and Wei (2002).     

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.     

Combining the unrestricted estimators into a single estimator and a simulation study on the unrestricted estimators

The purpose of this paper is to combine several regression estimators (ordinary least squares (OLS), ridge, contraction, principal components regression (PCR), Liu, r?k and r?d class estimators) into a single estimator. The conditions for the superiority of this new estimator over the PCR, the r?k class, the r?d class, β?(k, d), OLS, ridge, Liu and contraction estimators are derived by the scalar mean square error criterion and the estimators of the biasing parameters for this new estimator are examined. Also, a numerical example based on Hald data and a simulation study are used to illustrate the results.     

Ridge Regression – A Simulation Study

In this article we assess the suitability of two new ridge estimators by means of a simulation study. We compare these estimators with well-known ridge estimators. We also make direct comparisons between the ordinary least squares (OLS) estimator and the ridge estimators by using ratio of the average total mean square error of the OLS estimator and the ridge estimators. We find that the new estimators perform well under certain conditions.     

The relative efficiency of the restricted estimators in linear regression models

This paper deals with the problem of multicollinearity in a multiple linear regression model with linear equality restrictions. The restricted two parameter estimator which was proposed in case of multicollinearity satisfies the restrictions. The performance of the restricted two parameter estimator over the restricted least squares (RLS) estimator and the ordinary least squares (OLS) estimator is examined under the mean square error (MSE) matrix criterion when the restrictions are correct and not correct. The necessary and sufficient conditions for the restricted ridge regression, restricted Liu and restricted shrunken estimators, which are the special cases of the restricted two parameter estimator, to have a smaller MSE matrix than the RLS and the OLS estimators are derived when the restrictions hold true and do not hold true. Theoretical results are illustrated with numerical examples based on Webster, Gunst and Mason data and Gorman and Toman data. We conduct a final demonstration of the performance of the estimators by running a Monte Carlo simulation which shows that when the variance of the error term and the correlation between the explanatory variables are large, the restricted two parameter estimator performs better than the RLS estimator and the OLS estimator under the configurations examined.     

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.     

THE EXAMINATION AND ANALYSIS OF RESIDUALS FOR SOME BIASED ESTIMATORS IN LINEAR REGRESSION

In the presence of collinearity certain biased estimation procedures like ridge regression, generalized inverse estimator, principal component regression, Liu estimator, or improved ridge and Liu estimators are used to improve the ordinary least squares (OLS) estimates in the linear regression model. In this paper new biased estimator (Liu estimator), almost unbiased (improved) Liu estimator and their residuals will be analyzed and compared with OLS residuals in terms of mean-squared error.     

State space modeling of multiple time series: a comment

This paper provides guidance in choosing k₁ andk₂ of the double k-class (KK) estimator such that it will improve upon both the ordinary least squares (OLS) and Stein-rule (SR) estimators in predictive mean squared error (PMSE). Asymptotic bias and mean squared error (MSE) results are derived for nonnormal and other cases. A simulation compares the KK estimator with the OLS and SR estimators.     

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.     

MSE performance of the weighted average estimators consisting of shrinkage estimators when each individual regression coefficient is estimated

In this paper, we analytically derive the exact formula for the mean squared error (MSE) of two weighted average (WA) estimators for each individual regression coefficient. Further, we execute numerical evaluations to investigate small sample properties of the WA estimators, and compare the MSE performance of the WA estimators with the other shrinkage estimators and the usual OLS estimator. Our numerical results show that (1) the WA estimators have smaller MSE than the other shrinkage estimators and the OLS estimator over a wide region of parameter space; (2) the range where the relative MSE of the WA estimator is smaller than that of the OLS estimator gets narrower as the number of explanatory variables k increases.     

Exact small sample properties of an operational variant of the minimum mean squared error estimator

In this paper, we show a sufficient condition for an operational variant of the minimum mean squared error estimator (simply, the minimum MSE estimator) to dominate the ordinary least squares (OLS) estimator. It is also shown numerically that the minimum MSE estimator dominates the OLS estimator if the number of regression coefficients is larger than or equal to three, even if the sufficient condition is not satisfied. When the number of regression coefficients is smaller than three, our numerical results show that the gain in MSE of using the minimum MSE estimator is larger than the loss.     

Risk comparison of the Stein-rule estimator in a linear regression model with omitted relevant regressors and multivariate<Emphasis Type="Italic">t</Emphasis> errors under the Pitman nearness criterion

In this paper we consider a linear regression model with omitted relevant regressors and multivariatet error terms. The explicit formula for the Pitman nearness criterion of the Stein-rule (SR) estimator relative to the ordinary least squares (OLS) estimator is derived. It is shown numerically that the dominance of the SR estimator over the OLS estimator under the Pitman nearness criterion can be extended to the case of the multivariatet error distribution when the specification error is not severe. It is also shown that the dominance of the SR estimator over the OLS estimator cannot be extended to the case of the multivariatet error distribution when the specification error is severe. This research is partially supported by the Grants-in-Aid for 21st Century COE program.     

Nonparametric density estimation based on beta prime kernel

Abstract

In this work, we propose beta prime kernel estimator for estimation of a probability density functions defined with nonnegative support. For the proposed estimator, beta prime probability density function used as a kernel. It is free of boundary bias and nonnegative with a natural varying shape. We obtained the optimal rate of convergence for the mean squared error (MSE) and the mean integrated squared error (MISE). Also, we use adaptive Bayesian bandwidth selection method with Lindley approximation for heavy tailed distributions and compare its performance with the global least squares cross-validation bandwidth selection method. Simulation studies are performed to evaluate the average integrated squared error (ISE) of the proposed kernel estimator against some asymmetric competitors using Monte Carlo simulations. Moreover, real data sets are presented to illustrate the findings.     

Classification trees aided mixed regression model

This paper introduces a novel hybrid regression method (MixReg) combining two linear regression methods, ordinary least square (OLS) and least squares ratio (LSR) regression. LSR regression is a method to find the regression coefficients minimizing the sum of squared error rate while OLS minimizes the sum of squared error itself. The goal of this study is to combine two methods in a way that the proposed method superior both OLS and LSR regression methods in terms of R² statistics and relative error rate. Applications of MixReg, on both simulated and real data, show that MixReg method outperforms both OLS and LSR regression.     

A Local Linear Least-Absolute-Deviations Estimator of Volatility

An important empirical characteristic of financial time series is that the unconditional distribution of the returns tends to possess heavy tails. This is the motivation for the particular local kernel volatility estimator proposed in this work. Whereas least-square-deviations (LSD) estimators are strongly affected by heavy-tailed distributions, the performance of least-absolute-deviations (LAD) estimators is not. This robustness to heavy tails is evidenced by the more flexible assumptions made on the distributional moments of the observable variable. The simulation examples also highlight the superior performances of the LAD estimator when compared to the LSD estimator under heavy tails conditions. The full nonparametric model is described and the asymptotic properties of the LAD estimator are derived. Extensive Monte Carlo studies strongly suggest that the LAD estimator is asymptotically adaptive to the unknown conditional first moment. The LAD estimator is also used to estimate the volatility of the S&P500 and the BOVESPA returns.     

Finite sample properties of an HPT estimator when each individual regression coefficient is estimated in a misspecified linear regression model

ABSTRACT

In this paper, assuming that there exist omitted variables in the specified model, we analytically derive the exact formula for the mean squared error (MSE) of a heterogeneous pre-test (HPT) estimator whose components are the ordinary least squares (OLS) and feasible ridge regression (FRR) estimators. Since we cannot examine the MSE performance analytically, we execute numerical evaluations to investigate small sample properties of the HPT estimator, and compare the MSE performance of the HPT estimator with those of the FRR estimator and the usual OLS estimator. Our numerical results show that (1) the HPT estimator is more efficient when the model misspecification is severe; (2) the HPT estimator with the optimal critical value obtained under the correctly specified model can be safely used even when there exist omitted variables in the specified model.