Liu-type Estimator Research Articles

Leverage and influential observations on the Liu type estimator in the linear regression model with the severe collinearity

In the process of building a linear regression model, the essential part is to identify influential observations. Various influence measures involving Cook's distance and DFFITS are designed to detect the linear regression's influential observations using the Least Squares (LS). The existence of influential observations in the data is complicated by the presence of severe collinearity and affects the efficiency of the detection measures. This paper proposes new diagnostic methods based on the Liu type estimator (LTE) defined by Liu [1]. The Cook's distance and DFFITS for the LTE are introduced. Moreover, approximate formulas for Cook's distance and DFFITS are also proposed for LTE. Two real data sets with a high level of multicollinearity among the explanatory variables as well as the simulation study are used to illustrate and evaluate performance of the methodologies presented in this paper.

Robust Liu-Type Estimator for SUR Model

The Liu-type estimator is one of the shrink estimators that is used to remedy for a problem of multicollinearityin SUR model, but it is sensitive to the outlier. In this paper, we introduce the S Liu-type (SLiu-type) and MM Liu-type estimator (MMLiu-type) for SUR model. These estimators merge Liu-type estimator with S-estimator and with MM-estimator which makes it have high robustness at the different level of efficiency and at the same time prevents the bad effects of multicollinearity. Moreover, to get more robust features, we have modified the Liu-type estimator by making it depend on MM estimator instead of GLS estimator. The asymptotical properties for the suggested estimator were discussed and we used the fast and robust bootstrap (FRB) to obtain the suggested robust estimators. Furthermore, we run the simulation study to show the extent of excellence for the suggested robust estimators relative to the other estimators by many factors.

Some Shrinkage estimators based on median ranked set sampling

In this study, some shrinkage estimators using a median ranked set sample in the presence of multicollinearity were studied. Initially, we constructed the multiple regression model using median ranked set sampling. We also adapted the Ridge and Liu-type estimators to these multiple regression model. To investigate the efficiency of these estimators, a simulation study was performed for a different number of explanatory variables, sample sizes, correlation coefficients, and error variances in perfect and imperfect ranking cases. In addition, these estimators were compared with other estimators that are based on ranked set sample using simulation study. It is shown that when the collinearity is moderate, Ridge estimator using median ranked set sample performs better than other estimators and when the collinearity increases, Liu-type estimator using median ranked set sample gets better than all other estimators do. When the collinearity is smaller than 0.95, ridge estimator based on median ranked set sample is more efficient than Liu-type estimator based on same sample. However, this threshold increases as the sample size increases and the number of explanatory variables decreases. In addition, real data example is presented to illustrate how collinearity affects the estimators under median ranked set sampling and ranked set sampling.

The Liu-Type Estimator Based on Parameter Optimization and its Application in SBAS Deformation Model Inversion

A situation in which an image is combined with multiple images to form interferometric pairs is often observed in small baseline subset-interferometric synthetic aperture radar (SBAS-InSAR) deformation inversion, and this situation leads to a near linear correlation between the column vectors of the model design matrix. The Liu-type estimator introduces the parameters <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> into the normal equation to reduce the condition number of the design matrix and to improve the fitting properties. As the parameter <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> is mainly used to reduce ill-posed problems of the design matrix, the value of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> is not limited. However, the value of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> , as determined by existing methods, is usually too large or too small. Since the calculation of the mean square error involves true values, the parameter <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> is often affected by errors in the estimation results, which leads to the decreased accuracy of Liu-type estimation results. To determine the optimal value of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> , an iterative Liu-type estimator is proposed to eliminate errors. Then, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> -curve optimization method and iterative Liu-type estimator are combined to achieve the optimal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> . The reliability and accuracy of the methods are analyzed through SBAS-InSAR deformation experiments. The experimental results show that after using the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> -curve method and an iterative operation to optimize <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> , the accuracy of the Liu-type estimator based on parameter optimization is clearly improved compared with that of the ridge estimator and the Liu-type estimator.

On Liu-type biased estimators in measurement error models

ABSTRACT This paper considers the shrinkage estimation of parameters of measurement error models when it is suspected that the parameters may belong to a linear subspace. The class of Liu type estimators is proposed by choosing five quasi-empirical Bayes estimators in the presence of measurement errors in the data. This class of estimator combines the sample and prior information together along with the good properties of ridge estimators and chosen five quasi-empirical Bayes estimators. The advantages of the proposed class of estimators over the classical ridge regression estimator is that the quasi-empirical Bayes estimators are a linear function of the tuning parameter. When data has problems of measurement errors and multicollinearity, then these estimators can handle both the issues simultaneously. The asymptotic properties of the estimators are derived and analyzed. A Monte Carlo simulation is conducted and its findings are reported.

A new alternative estimation method for Liu-type logistic estimator via particle swarm optimization: an application to data of collapse of Turkish commercial banks during the Asian financial crisis

In the existence of multicollinearity problem in the logistic model, some important problems may occur in the analysis of the model, such as unstable maximum likelihood estimator with very high standard errors, false inferences. The Liu-type logistic estimator was proposed as two-parameter estimator to overcome multicollinearity problem in the logistic model. In the existing previous studies, the (k, d) pair in this shrinkage estimator is estimated by two-phase methods. However, since the different estimators can be utilized in the estimation of d, optimal choice of the (k, d) pair provided using the two-phase approaches is not guaranteed to overcome multicollinearity. In this article, a new alternative method based on particle swarm optimization is suggested to estimate (k, d) pair in Liu-type logistic estimator, simultaneously. For this purpose, an objective function that eliminates the multicollinearity problem, provides minimization of the bias of the model and improvement of the model’s predictive performance, is developed. Monte Carlo simulation study is conducted to show the performance of the proposed method by comparing it with existing methods. The performance of the proposed method is also demonstrated by the real dataset which is related to the collapse of commercial banks in Turkey during Asian financial crisis.

A new Liu-type estimator in binary logistic regression models

In logistic regression models, the maximum likelihood method is commonly used to estimate the model parameters. However, unstable parameter estimates are obtained as a result of multicollinearity. In this article, a new biased estimator is proposed to combat multicollinearity in the binary logistic regression models. The proposed estimator is a general estimator which includes other biased estimators, such as the Logistic Ridge, Logistic Liu and the estimators with two biasing parameters as special cases. Necessary and sufficient conditions for the superiority of the new biased estimator over the existing estimators are obtained. Also, Monte Carlo simulation studies are executed to compare the performance of the proposed biased estimator. Finally, a numerical example is given to illustrate some of the theoretical results.

A New Ridge-Type Estimator for the Linear Regression Model: Simulations and Applications.

The ridge regression-type (Hoerl and Kennard, 1970) and Liu-type (Liu, 1993) estimators are consistently attractive shrinkage methods to reduce the effects of multicollinearity for both linear and nonlinear regression models. This paper proposes a new estimator to solve the multicollinearity problem for the linear regression model. Theory and simulation results show that, under some conditions, it performs better than both Liu and ridge regression estimators in the smaller MSE sense. Two real-life (chemical and economic) data are analyzed to illustrate the findings of the paper.

New Versions of Liu-type Estimator in Weighted and non-weighted Mixed Regression Model

This paper considers and proposes new estimators that depend on the sample and on prior information in the case that they either are equally or are not equally important in the model. The prior information is described as linear stochastic restrictions. We study the properties and the performances of these estimators compared to other common estimators using the mean squared error as a criterion for the goodness of fit. A numerical example and a simulation study are proposed to explain the performance of the estimators.

A new Liu-type estimator for the Inverse Gaussian Regression Model

ABSTRACTThe Inverse Gaussian Regression Model (IGRM) is used when the response variable is positively skewed and follows the inverse Gaussian distribution. In this article, we propose a Liu-type estimator to combat multicollinearity in the IGRM. The variance of the Maximum Likelihood Estimator (MLE) is overstated due to the presence of severe multicollinearity. Moreover, some estimation methods are suggested to estimate the optimal value of the shrinkage parameter. The performance of the proposed estimator is compared with the MLE and some other existing estimators in the sense of mean squared error through Monte Carlo simulation and different real-life applications. Under certain conditions, it is concluded that the proposed estimator is superior to the MLE, ridge, and Liu estimator.

A stochastic restricted mixed Liu-type estimator in logistic regression model

A stochastic restricted mixed Liu-type estimator in logistic regression model

An improved and efficient biased estimation technique in logistic regression model

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 nonlinear goal programming model for determining the bias parameters of Liu-type estimator with application to construction sector in Egypt

A nonlinear goal programming model for determining the bias parameters of Liu-type estimator with application to construction sector in Egypt

Liu-type estimator for the gamma regression model

In this paper, we propose a new biased estimator called Liu-type estimator in gamma regression models in the presence of collinearity. We also consider some other estimators such as ridge estimator and Liu estimator and conduct a Monte Carlo simulation study to compare the estimators under different designs of collinearity. Moreover, we provide a real data application to show the applicability of the new estimator. The simulations and real data results show that the proposed estimator outperforms better than other competitor estimators by yielding smaller mean squared error (MSE).

Elliptical difference based ridge and Liu type estimators in partial linear measurement error models

In this article the difference based ridge and Liu type estimators are defined in partial linear models when the covariates are measured with additive errors. It is assumed that the additive error distributed according to the law belonging to the class of elliptically contoured distributions. The development of the corrected score function with the family of elliptical distributions is the basis for derivation of the estimators. The asymptotic normality of the estimates are established. Also, the conditions for the superiority of the proposed estimator over its counterpart, for selecting the biasing parameters are obtained. A simulation study is also performed.

Application of the Liu-type estimator in illposed problems of small baseline subsets deformation monitoring

To overcome the illposed problems existing in small baseline subsets deformation, the Liu-type estimator is used to improve the accuracy of calculation. The Liu-type estimator introduces parameter d to improve the fitting and statistical properties after the condition number of the design matrix is reduced to the desired level, which makes the solution of illposed problems robust. In the experiment, subsidence rates relevant to the Beijing area are computed from the ENVISAT satellite radar data from the European Space Agency from 2007 to 2010. The experimental results show that the accuracy of the deformation rates computed with the Liu-type estimator is clearly improved.

Liu-Type Multinomial Logistic Estimator

Multicollinearity in multinomial logistic regression affects negatively on the variance of the maximum likelihood estimator. That leads to inflated confidence intervals and theoretically important variables become insignificant in testing hypotheses. In this paper, Liu-type estimator is proposed that has smaller total mean squared error than the maximum likelihood estimator. The proposed estimator is a general estimator which includes other biased estimators such as Liu estimator and ridge estimator as special cases. Simulation studies and an application are given to evaluate the performance of our estimator. The results indicate that the proposed estimator is more efficient and reliable than the conventional estimators.

AN INVESTIGATION INTO PROPERTIES OF JACKKNIFED AND BOOTSTRAPPED LIU-TYPE ESTIMATOR

In 2003, Liu proposed a new estimator dealing with the problem of multicollinearity in linear regression model pointing out a drawback of ridge estimator used in this context. This new estimator, called Liu-type estimator was demonstrated to have lesser mean squared error than ridge estimator and ordinary least squares estimator, however, it may carry a large amount of bias. In the present paper, we propose different estimators in order to reduce the bias of Liu-type estimator, one using the Jackknife technique and other using the technique proposed in Kadiyala \cite{kad1984}. We also investigate the Bootstrap method of bias correction on the Liu-type estimator as well. The bias and mean squared error of these estimators have been compared using a simulation study as well as a numerical example. LSTA-2016-0059

On the stochastic restricted Liu-type maximum likelihood estimator in logistic regression model

In order to overcome multicollinearity, we propose a stochastic restricted Liu-type maximum likelihood estimator by incorporating Liu-type maximum likelihood estimator to the logistic regression model when the linear restrictions are stochastic. We also discuss the properties of the new estimator. Moreover, we give a method to choose the biasing parameter in the new estimator. Finally, a simulation study is given to show the performance of the new estimator.

Ortalama Karesel Hata Matrisi Kriterine göre Yanlı Tahmin Ediciler Üzerine Çalışma

Regression models with multicollinearity can be tackled by using various estimators such as class estimators, principal components regression, Liu-type estimators. In this study, we defined conditions where the class estimator is superior over the biased estimators in terms of mean square error matrix (MSEM) criterion. Finally, we showed theoretical results by means of a numerical example and a simulation study.