Fractional principal components regression: a general approach to biased estimators

Several biased estimators have been proposed as alternatives to the least squares estimator when multicollinearity is present in the multiple linear regression model. The ridge estimator and the principal components estimator are two techniques that have been proposed for such problems. In this paper the class of fractional principal component estimators is developed for the multiple linear regression model. This class contains many of the biased estimators commonly used to combat multicollinearity. In the fractional principal components framework, two new estimation techniques are introduced. The theoretical performances of the new estimators are evaluated and their small sample properties are compared via simulation with the ridge, generalized ridge and principal components estimators     

A modified ridge m-estimator for linear regression model with multicollinearity and outliers

The ordinary least-square estimators for linear regression analysis with multicollinearity and outliers lead to unfavorable results. In this article, we propose a new robust modified ridge M-estimator (MRME) based on M-estimator (ME) to deal with the combined problem resulting from multicollinearity and outliers in the y-direction. MRME outperforms modified ridge estimator, robust ridge estimator and ME, according to mean squares error criterion. Furthermore, a numerical example and a Monte Carlo simulation experiment are given to illustrate some of the theoretical results.     

A new modified Jackknifed estimator for the Poisson regression model

The Poisson regression is very popular in applied researches when analyzing the count data. However, multicollinearity problem arises for the Poisson regression model when the independent variables are highly intercorrelated. Shrinkage estimator is a commonly applied solution to the general problem caused by multicollinearity. Recently, the ridge regression (RR) estimators and some methods for estimating the ridge parameter k in the Poisson regression have been proposed. It has been found that some estimators are better than the commonly used maximum-likelihood (ML) estimator and some other RR estimators. In this study, the modified Jackknifed Poisson ridge regression (MJPR) estimator is proposed to remedy the multicollinearity. A simulation study and a real data example are provided to evaluate the performance of estimators. Both mean-squared error and the percentage relative error are considered as the performance criteria. The simulation study and the real data example results show that the proposed MJPR method outperforms the Poisson ridge regression, Jackknifed Poisson ridge regression and the ML in all of the different situations evaluated in this paper.     

Robust Liu estimator for regression based on an M-estimator

Consider the regression model y = beta 0 1 + Xbeta + epsilon. Recently, the Liu estimator, which is an alternative biased estimator beta L (d) = (X'X + I) -1 (X'X + dI)beta OLS , where 0<d<1 is a parameter, has been proposed to overcome multicollinearity . The advantage of beta L (d) over the ridge estimator beta R (k) is that beta L (d) is a linear function of d. Therefore, it is easier to choose d than to choose k in the ridge estimator. However, beta L (d) is obtained by shrinking the ordinary least squares (OLS) estimator using the matrix (X'X + I) -1 (X'X + dI) so that the presence of outliers in the y direction may affect the beta L (d) estimator. To cope with this combined problem of multicollinearity and outliers, we propose an alternative class of Liu-type M-estimators (LM-estimators) obtained by shrinking an M-estimator beta M , instead of the OLS estimator using the matrix (X'X + I) -1 (X'X + dI).     

On the restricted r–k class estimator and the restricted r–d class estimator in linear regression

In this article, the restricted r–k class estimator and restricted r–d class estimator are introduced, which are general estimators of the r–k class estimator by Baye and Parker [Combining ridge and principal component regression: A money demand illustration, Commun. Stat. Theory Methods 13(2) (1984), pp. 197–205] and the r–d class estimator by Kaç?ranlar and Sakall?o?lu [Combining the Liu estimator and the principal component regression estimator, Commun. Stat. Theory Methods 30(12) (2001), pp. 2699–2705], respectively. For the two cases when the restrictions are true and not true, the superiority of the restricted r–k class estimator and r–d class estimator over the restricted ridge regression estimator by Sarkar [A new estimator combining the ridge regression and the restricted least squares methods of estimation, Commun. Stat. Theory Methods 21 (1992), pp. 1987–2000] and the restricted Liu estimator by Kaç?ranlar et al. [A new biased estimator in linear regression and a detailed analysis of the widely analysed dataset on Portland cement, Sankhya - Indian J. Stat. 61B(3) (1999), pp. 443–459] are discussed with respect to the mean squared error matrix criterion. Furthermore, a Monte Carlo evaluation of the estimators is given to illustrate some of the theoretical results.     

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.     

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 influence function of penalized regression estimators

To perform regression analysis in high dimensions, lasso or ridge estimation are a common choice. However, it has been shown that these methods are not robust to outliers. Therefore, alternatives as penalized M-estimation or the sparse least trimmed squares (LTS) estimator have been proposed. The robustness of these regression methods can be measured with the influence function. It quantifies the effect of infinitesimal perturbations in the data. Furthermore, it can be used to compute the asymptotic variance and the mean-squared error (MSE). In this paper we compute the influence function, the asymptotic variance and the MSE for penalized M-estimators and the sparse LTS estimator. The asymptotic biasedness of the estimators make the calculations non-standard. We show that only M-estimators with a loss function with a bounded derivative are robust against regression outliers. In particular, the lasso has an unbounded influence function.     

Generalized preliminary test stochastic restricted estimator in the linear regression model

ABSTRACT

In this paper, we propose three generalized estimators, namely, generalized unrestricted estimator (GURE), generalized stochastic restricted estimator (GSRE), and generalized preliminary test stochastic restricted estimator (GPTSRE). The GURE can be used to represent the ridge estimator, almost unbiased ridge estimator (AURE), Liu estimator, and almost unbiased Liu estimator. When stochastic restrictions are available in addition to the sample information, the GSRE can be used to represent stochastic mixed ridge estimator, stochastic restricted Liu estimator, stochastic restricted almost unbiased ridge estimator, and stochastic restricted almost unbiased Liu estimator. The GPTSRE can be used to represent the preliminary test estimators based on mixed estimator. Using the GPTSRE, the properties of three other preliminary test estimators, namely preliminary test stochastic mixed ridge estimator, preliminary test stochastic restricted almost unbiased Liu estimator, and preliminary test stochastic restricted almost unbiased ridge estimator can also be discussed. The mean square error matrix criterion is used to obtain the superiority conditions to compare the estimators based on GPTSRE with some biased estimators for the two cases for which the stochastic restrictions are correct, and are not correct. Finally, a numerical example and a Monte Carlo simulation study are done to illustrate the theoretical findings of the proposed estimators.     

Improved robust ridge M-estimation

It is developed that non-sample prior information about regression vector-parameter, usually in the form of constraints, improves the risk performance of the ordinary least squares estimator (OLSE) when it is shrunken. However, in practice, it may happen that both multicollinearity and outliers exist simultaneously in the data. In such a situation, the use of robust ridge estimator is suggested to overcome the undesirable effects of the OLSE. In this article, some prior information in the form of constraints is employed to improve the performance of this estimator in the multiple regression model. In this regard, shrinkage ridge robust estimators are defined. Advantages of the proposed estimators over the usual robust ridge estimator are also investigated using Monte-Carlo simulation as well as a real data example.     

A robust Liu regression estimator

The least-squares regression estimator can be very sensitive in the presence of multicollinearity and outliers in the data. We introduce a new robust estimator based on the MM estimator. By considering weights, also the resulting MM-Liu estimator is highly robust, but also the estimation of the biasing parameter is robustified. Also for high-dimensional data, a robust Liu-type estimator is introduced, based on the Partial Robust M-estimator. Simulation experiments and a real dataset show the advantages over the standard estimators and other robustness proposals.     

Robust two parameter ridge M-estimator for linear regression

The problem of multicollinearity and outliers in the data set produce undesirable effects on the ordinary least squares estimator. Therefore, robust two parameter ridge estimation based on M-estimator (ME) is introduced to deal with multicollinearity and outliers in the y-direction. The proposed estimator outperforms ME, two parameter ridge estimator and robust ridge M-estimator according to mean square error criterion. Moreover, a numerical example and a Monte Carlo simulation experiment are presented.     

A new linearized ridge Poisson estimator in the presence of multicollinearity

Poisson regression is a very commonly used technique for modeling the count data in applied sciences, in which the model parameters are usually estimated by the maximum likelihood method. However, the presence of multicollinearity inflates the variance of maximum likelihood (ML) estimator and the estimated parameters give unstable results. In this article, a new linearized ridge Poisson estimator is introduced to deal with the problem of multicollinearity. Based on the asymptotic properties of ML estimator, the bias, covariance and mean squared error of the proposed estimator are obtained and the optimal choice of shrinkage parameter is derived. The performance of the existing estimators and proposed estimator is evaluated through Monte Carlo simulations and two real data applications. The results clearly reveal that the proposed estimator outperforms the existing estimators in the mean squared error sense.KEYWORDS: Poisson regression, multicollinearity, ridge Poisson estimator, linearized ridge regression estimator, mean squared errorMathematics Subject Classifications: 62J07, 62F10     

ROBUST RIDGE REGRESSION BASED ON AN M-ESTIMATOR

Consider the linear regression model y =β₀1 +Xβ+ in the usual notation. It is argued that the class of ordinary ridge estimators obtained by shrinking the least squares estimator by the matrix (X¹X + kI)^-1X'X is sensitive to outliers in the ^variable. To overcome this problem, we propose a new class of ridge-type M-estimators, obtained by shrinking an M-estimator (instead of the least squares estimator) by the same matrix. Since the optimal value of the ridge parameter k is unknown, we suggest a procedure for choosing it adaptively. In a reasonably large scale simulation study with a particular M-estimator, we found that if the conditions are such that the M-estimator is more efficient than the least squares estimator then the corresponding ridge-type M-estimator proposed here is better, in terms of a Mean Squared Error criteria, than the ordinary ridge estimator with k chosen suitably. An example illustrates that the estimators proposed here are less sensitive to outliers in the y-variable than ordinary ridge estimators.     

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.     

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.     

A new kind of stochastic restricted biased estimator for logistic regression model

In the logistic regression model, the variance of the maximum likelihood estimator is inflated and unstable when the multicollinearity exists in the data. There are several methods available in literature to overcome this problem. We propose a new stochastic restricted biased estimator. We study the statistical properties of the proposed estimator and compare its performance with some existing estimators in the sense of scalar mean squared criterion. An example and a simulation study are provided to illustrate the performance of the proposed estimator.KEYWORDS: Logistic regression, maximum likelihood estimator, mean squared error matrix, ridge regression, simulation study, stochastic restricted estimatorMathematics Subject Classifications: Primary 62J05, Secondary 62J07     

A graphical evaluation of logistic ridge estimator in mixture experiments

In comparison to other experimental studies, multicollinearity appears frequently in mixture experiments, a special study area of response surface methodology, due to the constraints on the components composing the mixture. In the analysis of mixture experiments by using a special generalized linear model, logistic regression model, multicollinearity causes precision problems in the maximum-likelihood logistic regression estimate. Therefore, effects due to multicollinearity can be reduced to a certain extent by using alternative approaches. One of these approaches is to use biased estimators for the estimation of the coefficients. In this paper, we suggest the use of logistic ridge regression (RR) estimator in the cases where there is multicollinearity during the analysis of mixture experiments using logistic regression. Also, for the selection of the biasing parameter, we use fraction of design space plots for evaluating the effect of the logistic RR estimator with respect to the scaled mean squared error of prediction. The suggested graphical approaches are illustrated on the tumor incidence data set.     

A restricted Liu estimator for binary regression models and its application to an applied demand system

In this article, we propose a restricted Liu regression estimator (RLRE) for estimating the parameter vector, β, in the presence of multicollinearity, when the dependent variable is binary and it is suspected that β may belong to a linear subspace defined by Rβ?=?r. First, we investigate the mean squared error (MSE) properties of the new estimator and compare them with those of the restricted maximum likelihood estimator (RMLE). Then we suggest some estimators of the shrinkage parameter, and a simulation study is conducted to compare the performance of the different estimators. Finally, we show the benefit of using RLRE instead of RMLE when estimating how changes in price affect consumer demand for a specific product.     

Developing a Liu estimator for the negative binomial regression model: method and application

This paper introduces a new shrinkage estimator for the negative binomial regression model that is a generalization of the estimator proposed for the linear regression model by Liu [A new class of biased estimate in linear regression, Comm. Stat. Theor. Meth. 22 (1993), pp. 393–402]. This shrinkage estimator is proposed in order to solve the problem of an inflated mean squared error of the classical maximum likelihood (ML) method in the presence of multicollinearity. Furthermore, the paper presents some methods of estimating the shrinkage parameter. By means of Monte Carlo simulations, it is shown that if the Liu estimator is applied with these shrinkage parameters, it always outperforms ML. The benefit of the new estimation method is also illustrated in an empirical application. Finally, based on the results from the simulation study and the empirical application, a recommendation regarding which estimator of the shrinkage parameter that should be used is given.