Optimal generalized logistic estimator

In this paper, we propose a new efficient estimator namely Optimal Generalized Logistic Estimator (OGLE) for estimating the parameter in a logistic regression model when there exists multicollinearity among explanatory variables. Asymptotic properties of the proposed estimator are also derived. The performance of the proposed estimator over the other existing estimators in respect of Scalar Mean Square Error criterion is examined by conducting a Monte Carlo simulation.     

[HDDA] sparse subspace constrained partial least squares

ABSTRACT

In this paper, we investigate the objective function and deflation process for sparse Partial Least Squares (PLS) regression with multiple components. While many have considered variations on the objective for sparse PLS, the deflation process for sparse PLS has not received as much attention. Our work highlights a flaw in the Statistically Inspired Modification of Partial Least Squares (SIMPLS) deflation method when applied in sparse PLS regression. We also consider the Nonlinear Iterative Partial Least Squares (NIPALS) deflation in sparse PLS regression. To remedy the flaw in the SIMPLS method, we propose a new sparse PLS method wherein the direction vectors are constrained to be sparse and lie in a chosen subspace. We give insight into this new PLS procedure and show through examples and simulation studies that the proposed technique can outperform alternative sparse PLS techniques in coefficient estimation. Moreover, our analysis reveals a simple renormalization step that can be used to improve the estimation of sparse PLS direction vectors generated using any convex relaxation method.     

Operational ridge regression estimators under the prediction goal

In this paper we study the Mean Square Error and Conditional Mean Forecasting of Operational Ordinary Ridge Estimator. We use the G( ) functions to provide both the exact and the approximate bias and Mean Square Error of ordinary ridge estimator (ORE), We show, among other things, that ORE dominates OLS up to a certain order of approximation under the conditional mean forecasting sense.     

A Novel Regression Approach: Least Squares Ratio

Regression Analysis (RA) is one of the frequently used tool for forecasting. The Ordinary Least Squares (OLS) Technique is the basic instrument of RA and there are many regression techniques based on OLS. This paper includes a new regression approach, called Least Squares Ratio (LSR), and comparison of OLS and LSR according to mean square errors of estimation of theoretical regression parameters (mse ß) and dependent value (mse y).     

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.     

A New Asymmetric Interaction Ridge (AIR) Regression Method

Despite that interaction terms are standard tools of regression analysis, the side effects of the inclusion of these terms in models estimated by ordinary least squares (OLS) are yet not fully penetrated. The inclusion of interaction effects induces multicollinearity problems since all non zero values are equal between the interaction term and the regressor. In this article, we propose a procedure to remedy this problem by the use of new ridge regression (RR) shrinkage parameters—which we call the asymmetric interaction ridge (AIR) regression method. By means of Monte Carlo simulations we evaluate both OLS and AIR using the mean square error (MSE) performance criterion. The result from the simulation study confirms our hypothesis that AIR always should be preferred to OLS since it has a lower estimated MSE. Moreover, the advantages of our new method are demonstrated in an empirical application where positive asymmetric price transmission effects are exposed for the mortgage interest rates of Handelsbanken Stadshypotek. It is observed that the mortgage interest rates increase more fully and rapidly to an increase in the bank's borrowing costs than to a decrease. This asymmetry is defined as positive asymmetric price transmission (APT).     

OLS与ML：回归模型两种参数估计方法的比较研究

最小二乘法(OLS)和最大似然法(ML)是回归模型参数估计的两种最重要的方法。 但二者有着明显的差别,本文就二者之间的有关差别进行比较。     

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.     

Some ridge regression estimators for the zero-inflated Poisson model

The zero-inflated Poisson regression model is commonly used when analyzing economic data that come in the form of non-negative integers since it accounts for excess zeros and overdispersion of the dependent variable. However, a problem often encountered when analyzing economic data that has not been addressed for this model is multicollinearity. This paper proposes ridge regression (RR) estimators and some methods for estimating the ridge parameter k for a non-negative model. A simulation study has been conducted to compare the performance of the estimators. Both mean squared error and mean absolute error are considered as the performance criteria. The simulation study shows that some estimators are better than the commonly used maximum-likelihood estimator and some other RR estimators. Based on the simulation study and an empirical application, some useful estimators are recommended for practitioners.     

NONNEGATIVE ESTIMATORS FOR THE ONE-WAY RANDOM EFFECTS MODEL

The relative merits of ten estimators for the variance component of the balanced and unbalanced one-way random effects models are compared. Six of the estimators are nonnegative, two of which are obtained by modifying the Minimum Variance Quadratic Unbiased Estimator (MIVQUE) and the Weighted Least Square Estimator (WLS), and two more from the positive parts of these estimators. The Minimum Norm Quadratic Estimator (MINQE), which is nonnegative, is adjusted for reducing its bias. The nonnegative Minimum Mean Square Error Estimator (MIMSQE), the Analysis of Variance (ANOVA) and Unweighted Sums of Squares (USS) estimator are also included.     

Asymptotic confidence intervals in ridge regression based on the Edgeworth expansion

Ridge Regression techniques have been found useful to reduce mean square errors of parameter estimates when multicollinearity is present. But the usefulness of the method rest not only upon its ability to produce good parameter estimates, with smaller mean squared error than Ordinary Least Squares, but also on having reasonable inferential procedures. The aim of this paper is to develop asymptotic confidence intervals for the model parameters based on Ridge Regression estimates and the Edgeworth expansion. Some simulation experiments are carried out to compare these confidence intervals with those obtained from the application of Ordinary Least Squares. Also, an example will be provided based on the well known data set of Hald.     

A simulation study on SPSS ridge regression and ordinary least squares regression procedures for multicollinearity data

This study compares the SPSS ordinary least squares (OLS) regression and ridge regression procedures in dealing with multicollinearity data. The LS regression method is one of the most frequently applied statistical procedures in application. It is well documented that the LS method is extremely unreliable in parameter estimation while the independent variables are dependent (multicollinearity problem). The Ridge Regression procedure deals with the multicollinearity problem by introducing a small bias in the parameter estimation. The application of Ridge Regression involves the selection of a bias parameter and it is not clear if it works better in applications. This study uses a Monte Carlo method to compare the results of OLS procedure with the Ridge Regression procedure in SPSS.     

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.     

On the asymptotic bias of OLS in dynamic regression models with autocorrelated errors

It is well-known that Ordinary Least Squares (OLS) yields inconsistent estimates if applied to a regression equation with lagged dependent variables and correlated errors. Bias expressions which appear in the literature usually assume the exogenous variables to be non-stochastic. Due to this assumption the numerical sizes of these expressions cannot be determined. Further, the analysis is mostly restricted to very simple models. In this paper the problem of calculating the asymptotic bias of OLS is generalized to stationary dynamic regression models, where the errors follow a stationary ARMA process. A general bias expression is derived and a method is introduced by which its actual size can be computed numerically.     

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.     

Efficiency analysis of ten estimation procedures for quantitative linear models with autocorrelated errors

Many estimation procedures for quantitative linear models with autocorrelated errors have been proposed in the literature. A number of these procedures have been compared in various ways for different sample sizes and autocorrelation parameters values and for structured or random explanatory vaiables. In this paper, we revisit three situations that were considered to some extent in previous studies, by comparing ten estimation procedures: Ordinary Least Squares (OLS), Generalized Least Squares (GLS), estimated Generalized Least Squares (six procedures), Maximum Likelihood (ML), and First Differences (FD). The six estimated GLS procedures and the ML procedure differ in the way the error autocovariance matrix is estimated. The three situations can be defined as follows: Case 1, the explanatory variable x in the simple linear regression is fixed; Case 2,x is purely random; and Case 3x is first-order autoregressive. Following a theoretical presentation, the ten estimation procedures are compared in a Monte Carlo study conducted in the time domain, where the errors are first-order autoregressive in Cases 1-3. The measure of comparison for the estimation procedures is their efficiency relative to OLS. It is evaluated as a function of the time series length and the magnitude and sign of the error autocorrelation parameter. Overall, knowledge of the model of the time series process generating the errors enhances efficiency in estimated GLS. Differences in the efficiency of estimation procedures between Case 1 and Cases 2 and 3 as well as differences in efficiency among procedures in a given situation are observed and discussed.     

On Ridge Parameters in Logistic Regression

This article applies and investigates a number of logistic ridge regression (RR) parameters that are estimable by using the maximum likelihood (ML) method. By conducting an extensive Monte Carlo study, the performances of ML and logistic RR are investigated in the presence of multicollinearity and under different conditions. The simulation study evaluates a number of methods of estimating the RR parameter k that has recently been developed for use in linear regression analysis. The results from the simulation study show that there is at least one RR estimator that has a lower mean squared error (MSE) than the ML method for all the different evaluated situations.     

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).