Liu-type Estimator Research Articles

Testing the Significance of Regression Coefficients in Liu Type Estimators

In the linear regression model, the multicollinearity problem arises when there is a linear relationship between independent variables. This situation causes the variance of the estimations of the model parameters obtained by the Least Squares Estimator method to increase and move away from the true value, resulting in unstable and incorrect results. Biased Estimator methods are developed to eliminate the adverse effects caused by multicollinearity. In this study, a test statistic is obtained to test the significance of the model coefficients for the Liu-Type Estimator using the test statistic method suggested in the study of Halawa and El-Bassiouni (2000). With a simulation study, the significance of the model coefficients of the Ridge, Liu, and Liu type biased estimators in different situations is tested; the type I errors and power values of the estimators are calculated; the results are compared. In addition, a real data application is performed to better understand the test procedure.

Reducing Bias in Beta Regression Models Using Jackknifed Liu-Type Estimators: Applications to Chemical Data

In the field of chemical data modeling, it is common to encounter response variables that are constrained to the interval (0, 1). In such cases, the beta regression model is often a more suitable choice for modeling. However, like any regression model, collinearity can present a significant challenge. To address this issue, the Liu-type estimator has been used as an alternative to the maximum likelihood estimator, but it suffers from bias. In this paper, we introduce the Jackknifed Liu-type estimator and its modified version, which demonstrate improved bias reduction compared to the original Liu-type estimator. We assess the theoretical and numerical performance of these estimators through Monte Carlo simulations and real-data examples from the field of chemistry. Our findings highlight the significant improvements offered by the proposed estimators in terms of accuracy and reliability.

A two-parameter estimator for correlated regressors in gamma regression model

Gamma Modified Two Parameter (GMTP) is a novel biased two-parameter estimator proposed to address the effects of multicollinearity in Generalized Linear Models (GLMs). An expansion of the linear regression model's Modified Two Parameters (MTP) is the newly suggested estimator. The performance of the GMTP estimator over the maximum likelihood estimator (MLE), gamma ridge estimator (GRE), gamma Liu estimator (GLE), and gamma Liu-type estimator (GLTE) reviewed in this article are theoretically compared, and the estimator's properties is discussed. A simulation study that examine the effects of the dispersion parameter, sample size, explanatory variables, and degree of correlation are used to examine the superiority of the GMTP with four different biasing parameters over the MLE, GRE, GLE, and GLTE with regard to the estimated MSE criterion. The GMTP estimator with biasing parameters and outperforms the MLE, GRE, GLE, and GLTE, according to simulation research. More research can be done to see how well the GMTP estimator performs in comparison to other estimators that were not examined in this study. 2 k 4 k

Liu-type estimator in Conway–Maxwell–Poisson regression model: theory, simulation and application

Recently, many authors have been motivated to propose a new regression estimator in the case of multicollinearity. The most well-known of these estimators are ridge, Liu and Liu-type estimators. Many studies on regression models have shown that the Liu-type estimator is a good alternative to the ridge and Liu estimators in the literature. We consider a new Liu-type estimator, an alternative to ridge and Liu estimators in Conway–Maxwell–Poisson regression model. Moreover, we study the theoretical properties of the Liu-type estimator, and we provide some theorems showing under which conditions that the Liu-type estimator is superior to the others. Since there are two parameters of the Liu-type estimator, we also propose a method to select the parameters. We designed a simulation study to demonstrate the superiority of the Liu-type estimator compared to the ridge and Liu estimators. We also evaluated the usefulness and superiority of the proposed regression estimator with a practical data example. As a result of the simulation and real-world data example, we conclude that the proposed regression estimator is superior to its competitors according to the mean square error criterion.

Almost unbiased Liu-type estimator for Tobit regression and its application

This paper introduces a new estimator for the Tobit regression model when the multicollinearity exists. This estimator is called almost unbiased Tobit Liu-Type estimator, which is obtained by a mixture between the Liu-Type estimator and almost unbiased estimator. This estimator avoids the negative effects of multicollinearity on the maximum likelihood estimator. Furthermore, we check the superiority of the new estimator over the maximum likelihood estimator, Liu-Type estimator and almost unbiased estimator according to the simulated mean square error SMSE criteria. In addition, we run the simulation study to investigate the effects of a group of factors on the performance of the new estimator. Finally, to check the benefits of the new estimator via the real data, we use two applications for the Tobit regression, the English Premier League data and Egyptian agricultural GDP data.

A new improvement Liu-type estimator for the Bell regression model

The Poisson Regression Model (PRM) is a well-known model in applications when the response variable consists of count data. However, Bell Regression Model (BRM) is proposed recently as an alternative to the PRM in some cases where the data is over-dispersed. But, multicollinearity between explanatory variables negatively affects traditional estimation methods, such as MLE. Therefore, to avoid this problem, several shrinkage estimators are proposed in the BRM. In this study, a new improved Liu-type estimator is proposed as an alternative to the other proposed biased estimators for the BRM to model count data with over-dispersion. Furthermore, the Monte Carlo simulation studies are executed to compare the performances of the proposed biased estimators. Finally, the obtained results are illustrated in real data.

A new Liu-type estimator for the gamma regression model

In many real-world problems, there are situations where the dependent variable may have a Gamma distribution. The Gamma Regression Models (GRMs) are preferred when the response variable assumes a Gamma distribution with a given set of independent variables. The Maximum Likelihood Estimator (MLE) is used to estimate the unknown parameters. In the presence of multicollinearity, the variance of the MLE becomes inflated and the inference based on the MLE may not be reasonable. In this article, we propose a new biased estimator called the new Liu-type estimator in the GRMs to combat multicollinearity. The proposed estimator is a general estimator which includes other biased estimators, such as the Gamma ridge estimator, Gamma Liu estimator, and the estimators with two biasing parameters as special cases. Furthermore, several methods are proposed to determine the biasing parameters in the estimators. Also, a Monte Carlo simulation study has been conducted to assess the performance of the proposed biased estimator where the Estimated Mean Squared Error (EMSE) is considered as a performance criterion. Finally, two numerical examples are given to investigate the performance of the proposed estimator over existing estimators.

The beta Liu-type estimator: simulation and application

The Beta Regression Model (BRM) is commonly used while analyzing data where the dependent variable is restricted to the interval $[0,1]$ for example proportion or probability. The Maximum Likelihood Estimator (MLE) is used to estimate the regression coefficients of BRMs. But in the presence of multicollinearity, MLE is very sensitive to high correlation among the explanatory variables. For this reason, we introduce a new biased estimator called the Beta Liu-Type Estimator (BLTE) to overcome the multicollinearity problem in the case that dependent variable follows a Beta distribution. The proposed estimator is a general estimator which includes other biased estimators, such as the Ridge Estimator, Liu Estimator, and the estimators with two biasing parameters as special cases in BRM. The performance of the proposed new estimator is compared to the MLE and other biased estimators in terms of the Estimated Mean Squared Error (EMSE) criterion by conducting a simulation study. Finally, a numerical example is given to show the benefit of the proposed estimator over existing estimators.

Liu-type pretest and shrinkage estimation for the conditional autoregressive model.

Spatial regression models have recently received a lot of attention in a variety of fields to address the spatial autocorrelation effect. One important class of spatial models is the Conditional Autoregressive (CA). Theses models have been widely used to analyze spatial data in various areas, as geography, epidemiology, disease surveillance, civilian planning, mapping of poorness signals and others. In this article, we propose the Liu-type pretest, shrinkage and positive shrinkages estimators for the large-scale effect parameter vector of the CA regression model. The set of the proposed estimators are evaluated analytically via their asymptotic bias, quadratic bias, the asymptotic quadratic risks, and numerically via their relative mean squared errors. Our results demonstrate that the proposed estimators are more efficient than Liu-type estimator. To conclude this paper, we apply the proposed estimators to the Boston housing prices data, and applied a bootstrapping technique to evaluate the estimators based on their mean squared prediction error.

A new class of Poisson Ridge-type estimator

The Poisson Regression Model (PRM) is one of the benchmark models when analyzing the count data. The Maximum Likelihood Estimator (MLE) is used to estimate the model parameters in PRMs. However, the MLE may suffer from various drawbacks that arise due to the existence of multicollinearity problems. Many estimators have been proposed as alternatives to each other to alleviate the multicollinearity problem in PRM, such as Poisson Ridge Estimator (PRE), Poisson Liu Estimator (PLE), Poisson Liu-type Estimator (PLTE), and Improvement Liu-Type Estimator (ILTE). In this study, we define a new general class of estimators which is based on the PRE as an alternative to other existing biased estimators in the PRMs. The superiority of the proposed biased estimator over the other existing biased estimators is given under the asymptotic matrix mean square error sense. Furthermore, two separate Monte Carlo simulation studies are implemented to compare the performances of the proposed biased estimators. Finally, the performances of all considered biased estimators are shown in real data.

Coefficients Estimation of Linear Regression Models Using Liu-Type Shrinkage Estimators

This paper suggests Liu-type shrinkage estimators in linear regression model in the presence of multicollinearity under subspace information. The performance of the proposed estimators is compared to Liu-type estimator in terms of their relative efficiency via a Monte Carlo simulation study and a real data set. The results reveal that the proposed estimators outperform better than the Liu-type estimator.

K-L Estimator: Dealing with Multicollinearity in the Logistic Regression Model

Multicollinearity negatively affects the efficiency of the maximum likelihood estimator (MLE) in both the linear and generalized linear models. The Kibria and Lukman estimator (KLE) was developed as an alternative to the MLE to handle multicollinearity for the linear regression model. In this study, we proposed the Logistic Kibria-Lukman estimator (LKLE) to handle multicollinearity for the logistic regression model. We theoretically established the superiority condition of this new estimator over the MLE, the logistic ridge estimator (LRE), the logistic Liu estimator (LLE), the logistic Liu-type estimator (LLTE) and the logistic two-parameter estimator (LTPE) using the mean squared error criteria. The theoretical conditions were validated using a real-life dataset, and the results showed that the conditions were satisfied. Finally, a simulation and the real-life results showed that the new estimator outperformed the other considered estimators. However, the performance of the estimators was contingent on the adopted shrinkage parameter estimators.

A generalized Liu-type estimator for logistic partial linear regression model with multicollinearity

<abstract><p>This paper is concerned with proposing a generalized Liu-type estimator (GLTE) to address the multicollinearity problem of explanatory variable of the linear part in the logistic partially linear regression model. Using the profile likelihood method, we propose the GLTE as a general class of Liu-type estimator, which includes the profile likelihood estimator, the ridge estimator, the Liu estimator and the Liu-type estimator as special cases. The conditional superiority of the proposed GLTE over the other estimators is derived under the asymptotic mean square error matrix (MSEM) criterion. Moreover, the optimal choices of biasing parameters and function of biasing parameter are given. Numerical simulations demonstrate that the proposed GLTE performs better than the existing estimators. An application on a set of real data arising from the study of Indian Liver Patient is shown for illustrating our theoretical results.</p></abstract>

A new improved Liu-type estimator for Poisson regression models

The Poisson Regression Model (PRM) is commonly used in applied sciences such as economics and the social sciences when analyzing the count data. The maximum likelihood method is the well-known estimation technique to estimate the parameters in PRM. However, when the explanatory variables are highly intercorrelated, unstable parameter estimates can be obtained. Therefore, biased estimators are widely used to alleviate the undesirable effects of these problems. In this study, a new improved Liu-type estimator is proposed as an alternative to the other proposed biased estimators. The superiority of the new proposed estimator over the existing biased estimators is given under the asymptotic matrix mean square error criterion. Furthermore, Monte Carlo simulation studies are executed to compare the performances of the proposed biased estimators. Finally, the obtained results are illustrated in real data. Based on the set of experimental conditions which are investigated, the proposed biased estimator outperforms the other biased estimators.

Liu-type shrinkage strategies in zero-inflated negative binomial models with application to Expenditure and Default Data

In modeling count data with overdispersion and extra zeros, zero-inflated negative binomial (ZINB) regression model is useful. In a regression model, the multicollinearity problem arises when there are some high correlations between predictor variables. This problem leads to the maximum likelihood method will not be an efficient estimator. The ridge and Liu-type estimators have been proposed to combat the multicollinearity problem so that the Liu-type estimator is better. In this paper, we proposed the Liu-type shrinkage estimators, namely linear shrinkage, preliminary test, shrinkage preliminary test, Stein-type, and positive Stein-type Liu estimators to estimate the count parameters in the ZINB model, when some of the predictor variables have not a significant effect to predict the response variable so that a sub-model may be sufficient. The asymptotic distributional biases and variances of the proposed estimators are nicely demonstrated. We also compared the performance of the Liu-type shrinkage estimators along with the Liu-type unrestricted estimator by using an extensive Monte Carlo simulation study. The results show that the performances of the proposed estimators are superior to those based on Liu-type unrestricted estimators. We also applied the proposed estimation methods to Expenditure and Default Data.

Shifted Liu-Type Estimator in The Linear Regression

The methods to solve the problem of multicollinearity have an important issue in the linear regression. The Liu-type estimator is one of these methods used to reduce its effect. This estimator is an estimator with two parameters denoted and . Kurnaz and Akay (2015) [6] introduced a new approach for the Liu-type estimator and called it new Liu-type (NL) estimator. This proposed estimator is based on a continuous function of rather than two parameters and includes OLS, ridge estimator, Liu estimator, and some estimators with two biasing parameters as special cases. This study aimed to improve the NL estimator by shifting. The performance of the shifted NL estimator is compared to the NL estimator and other estimators depending on the mean squared error (MSE) criterion. The real data example and simulation study reveal that the SNL estimator can be a good selection in the linear regression model.

An almost unbiased Liu-type estimator in the linear regression model

A biased estimator, compared to least squares estimators, is one of the most used statistical procedures to overcome the problem of multicollinearity. Liu-type estimators, which are biased estimators, are preferred in a wide range of fields. In this article, we propose an almost unbiased Liu-type (AUNL) estimator and discuss its performance under the mean square error matrix criterion among existing estimators. The proposed AUNL estimator is a general estimator and is based on the function of a single biasing parameter. It includes an ordinary least squares estimator, an almost unbiased ridge estimator, an almost unbiased Liu estimator, and an almost unbiased two-parameter estimator. Finally, real data examples and a Monte Carlo simulation are provided to illustrate the theoretical results.

A New Two-Parameter Estimator for Beta Regression Model: Method, Simulation, and Application

The beta regression is a widely known statistical model when the response (or the dependent) variable has the form of fractions or percentages. In most of the situations in beta regression, the explanatory variables are related to each other which is commonly known as the multicollinearity problem. It is well-known that the multicollinearity problem affects severely the variance of maximum likelihood (ML) estimates. In this article, we developed a new biased estimator (called a two-parameter estimator) for the beta regression model to handle this problem and decrease the variance of the estimation. The properties of the proposed estimator are derived. Furthermore, the performance of the proposed estimator is compared with the ML estimator and other common biased (ridge, Liu, and Liu-type) estimators depending on the mean squared error criterion by making a Monte Carlo simulation study and through two real data applications. The results of the simulation and applications indicated that the proposed estimator outperformed ML, ridge, Liu, and Liu-type estimators.

Almost unbiased Liu-type estimators in gamma regression model

The Liu-type estimator has been consistently demonstrated to be an attractive shrinkage method to reduce the effect of multicollinearity problem. It is known that multicollinearity affects the variance of the maximum likelihood estimator negatively in gamma regression model. Therefore, an almost unbiased Liu-type estimator together with a modified version of it is proposed to overcome the multicollinearity problem. The performance of the new estimators is investigated both theoretically and numerically via a Monte Carlo simulation experiment and a real data illustration. Based on the results, it is observed that the proposed estimators can bring significant improvement relative to other competitor estimators.

Inverse Gaussian Liu-type estimator

The inverse Gaussian regression (IGR) model parameters are generally estimated using the maximum likelihood (ML) estimation method. Since the multicollinearity problem exists among the explanatory variables, the ML estimation method becomes inflated. When the multicollinearity problem occurs, biased estimators can be used to estimate the parameters of the model. One of the most widely used biased estimators is the Liu-type estimator. In this study, we extend the Liu-type estimator for the IGR model. The proposed estimator is compared with the Ridge and Liu estimators defined for the IGR model in terms of the mean squared error (MSE) criterion. Also, a real data example is presented to illustrate the superiority of the proposed estimator. Experimental results show that the Liu-type estimator outperforms the Ridge and Liu estimators when multicollinearity exists.