An improved and efficient biased estimation technique in logistic regression model

Abstract

In this article, we propose a new improved and efficient biased estimation method which is a modified restricted Liu-type estimator satisfying some sub-space linear restrictions in the binary logistic regression model. We study the properties of the new estimator under the mean squared error matrix criterion and our results show that under certain conditions the new estimator is superior to some other estimators. Moreover, a Monte Carlo simulation study is conducted to show the performance of the new estimator in the simulated mean squared error and predictive median squared errors sense. Finally, a real application is considered.     

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.     

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.     

Liu-Type Logistic Estimator

It is known that multicollinearity inflates the variance of the maximum likelihood estimator in logistic regression. Especially, if the primary interest is in the coefficients, the impact of collinearity can be very serious. To deal with collinearity, a ridge estimator was proposed by Schaefer et al. The primary interest of this article is to introduce a Liu-type estimator that had a smaller total mean squared error (MSE) than the Schaefer's ridge estimator under certain conditions. Simulation studies were conducted that evaluated the performance of this estimator. Furthermore, the proposed estimator was applied to a real-life dataset.     

On the weighted mixed Liu-type estimator under unbiased stochastic restrictions

In this article, we introduce the weighted mixed Liu-type estimator (WMLTE) based on the weighted mixed and Liu-type estimator (LTE) in linear regression model. We will also present necessary and sufficient conditions for superiority of the weighted mixed Liu-type estimator over the weighted mixed estimator (WME) and Liu type estimator (LTE) in terms of mean square error matrix (MSEM) criterion. Finally, a numerical example and a Monte Carlo simulation is also given to show the theoretical results.     

The distribution of the Liu-type estimator of the biasing parameter in elliptically contoured models

We derive the density function of the stochastic shrinkage parameters of the Liu-type estimator in elliptical models. The correctness of derivation is checked by simulations. A real data application is also provided.     

Liu-type estimator in semiparametric regression models

In this paper, we introduced a Liu-type estimator for the vector of parameters β in a semiparametric regression model. We also obtained the semiparametric restricted Liu-type estimator for the parametric component in a semiparametric regression model. The ideas in the paper are illustrated in a real data example and in a Monte Carlo simulation study.     

On the performance of the Jackknifed Liu-type estimator in linear regression model

In this paper, we are proposing a modified jackknife Liu-type estimator (MJLTE) that was created by combining the ideas underlying both the Liu-type estimator (LTE) and the jackknifed Liu-type estimator (JLTE). We will also present the necessary and sufficient conditions for superiority of the MJLTE over the LTE and JLTE, in terms of mean square error matrix criterion. Finally, a real data example and a Monte Carlo simulation are also given to illustrate theoretical results.     

On the Principal Component Liu-type Estimator in Linear Regression

In this article, we present a principal component Liu-type estimator (LTE) by combining the principal component regression (PCR) and LTE to deal with the multicollinearity problem. The superiority of the new estimator over the PCR estimator, the ordinary least squares estimator (OLSE) and the LTE are studied under the mean squared error matrix. The selection of the tuning parameter in the proposed estimator is also discussed. Finally, a numerical example is given to explain our theoretical results.     

Adjustive Liu-Type Estimators in Linear Regression Models

In this article, we aim to put forward the notion of adjustive Liu-type estimator (ALTE) in the linear regression model. First, the explicit expression of the optimal selection of the adjustive factors is derived under the PRESS criterion through matrix techniques. Then, the results are applied to the dataset on Portland cement. Moreover, to select biasing parameters from the theoretical point of view, we extend ALTE to the generalized version (GALTE) and obtained the optimal ones. The results of the Portland cement data show that ALTE's and GALTE's can substantially improve the ordinary least squares estimator and Liu-type estimators.     

Efficiency of two classes of stochastic restricted almost unbiased type principal component estimators in linear regression model

In this paper, we introduce two new classes of estimators called the stochastic restricted almost unbiased ridge-type principal component estimator (SRAURPCE) and the stochastic restricted almost unbiased Liu-type principal component estimator (SRAURPCE) to overcome the well-known multicollinearity problem in linear regression model. For the two cases when the restrictions are true and not true, necessary and sufficient conditions for the superiority of the proposed estimators are derived and compared, respectively. Furthermore, a Monte Carlo simulation study and a numerical example are given to illustrate the performance of the proposed estimators.     

Performance of the difference-based Liu-type estimator in partially linear model

This paper discusses the parameter estimation in a partially linear model. We proposed a difference-based Liu-type estimator of the unknown parameters in the partially linear model. The asymptotically properties of the proposed estimator are discussed. We propose a iterative method to choose the biasing parameters. Finally, a simulation study and a numerical example are presented to explain the performance of the estimators.     

A New Estimator for Cox Proportional Hazard Regression Model in Presence of Collinearity

We propose a new approach to estimate the parameters of the Cox proportional hazards model in the presence of collinearity. Generally, a maximum partial likelihood estimator is used to estimate parameters for the Cox proportional hazards model. However, the maximum partial likelihood estimators can be seriously affected by the presence of collinearity since the parameter estimates result in large variances.

In this study, we develop a Liu-type estimator for Cox proportional hazards model parameters and compare it with a ridge regression estimator based on the scalar mean squared error (MSE). Finally, we evaluate its performance through a simulation study.     

A test statistic to choose between Liu-type and least-squares estimator based on mean square error criteria

In this study, the necessary and sufficient conditions for the Liu-type (LT) biased estimator are determined. A test for choosing between the LT estimator and least-squares estimator is obtained by using these necessary and sufficient conditions. Also, a simulation study is carried out to compare this estimator against the ridge estimator. Furthermore, a numerical example is given for defined test statistic.     

Particle swarm optimization based Liu-type estimator

In this study, a new method for the estimation of the shrinkage and biasing parameters of Liu-type estimator is proposed. Because k is kept constant and d is optimized in Liu’s method, a (k, d) pair is not guaranteed to be the optimal point in terms of the mean square error of the parameters. The optimum (k, d) pair that minimizes the mean square error, which is a function of the parameters k and d, should be estimated through a simultaneous optimization process rather than through a two-stage process. In this study, by utilizing a different objective function, the parameters k and d are optimized simultaneously with the particle swarm optimization technique.     

Robust Liu-type estimator for regression based on M-estimator

The problem of multicollinearity and outliers in the dataset can strongly distort ordinary least-square estimates and lead to unreliable results. We propose a new Robust Liu-type M-estimator to cope with this combined problem of multicollinearity and outliers in the y-direction. Our new estimator has advantages over two-parameter Liu-type estimator, Ridge-type M-estimator, and M-estimator. Furthermore, we give a numerical example and a simulation study to illustrate some of the theoretical results.     

Optimal Estimator for Logistic Model with Distribution‐free Random Intercept

Logistic models with a random intercept are prevalent in medical and social research where clustered and longitudinal data are often collected. Traditionally, the random intercept in these models is assumed to follow some parametric distribution such as the normal distribution. However, such an assumption inevitably raises concerns about model misspecification and misleading inference conclusions, especially when there is dependence between the random intercept and model covariates. To protect against such issues, we use a semiparametric approach to develop a computationally simple and consistent estimator where the random intercept is distribution‐free. The estimator is revealed to be optimal and achieve the efficiency bound without the need to postulate or estimate any latent variable distributions. We further characterize other general mixed models where such an optimal estimator exists.     

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.     

A new estimator combining the ridge regression and the restricted least squares methods of estimation

It is well-known in the literature on multicollinearity that one of the major consequences of multicollinearity on the ordinary least squares estimator is that the estimator produces large sampling variances, which in turn might inappropriately lead to exclusion of otherwise significant coefficients from the model. To circumvent this problem, two accepted estimation procedures which are often suggested are the restricted least squares method and the ridge regression method. While the former leads to a reduction in the sampling variance of the estimator, the later ensures a smaller mean square error value for the estimator. In this paper we have proposed a new estimator which is based on a criterion that combines the ideas underlying these two estimators. The standard properties of this new estimator have been studied in the paper. It has also been shown that this estimator is superior to both the restricted least squares as well as the ordinary ridge regression estimators by the criterion of mean sauare error of the estimator of the regression coefficients when the restrictions are indeed correct. The conditions for superiority of this estimator over the other two have also been derived for the situation when the restrictions are not correct.     

An efficiency comparison of dual ratio and product estimatiors

The present paper investigates the efficiency of the dual ratio estimator under the super population model with uncorrelated errors and a gamma-distributed auxiliary variabble. It is found that the dual ratio estimator is more efficient than the product estimator when the auxiliary variable has a gamma distribution with parameter greater than or equal to one, in the case when the regression is through the origin or when the product of intercept and slope is positive.