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

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ZANDI Z.; BEVRANI H.

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Journal Paper

Paper Information

Journal: JOURNAL OF STATISTICAL SCIENCES Year:2023 | Volume:16 | Issue:2 Page(s): 417-434

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Information Journal Paper

Title

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

Author(s)

ZANDI Z. | BEVRANI H. | Issue Writer Certificate

Keywords

Linear regression model

Multicollinearity

Shrinkage estimators

Monte Carlo simulation

Abstract

Introduction In this study, we addressed parameter estimation in the Linear regression model in the presence of Multicollinearity when there exists some prior information about predictor variables that appears as a linear restriction on the model parameters. We estimated the parameters based on Liu-type linear shrinkage, preliminary test, Stein, and positive Stein strategies. The performance of the proposed estimators is compared to the Liu-type estimator in terms of their relative efficiency via a Monte Carlo simulation study and an actual data set. Material and Methods In the Linear regression model, the ordinary least squares (OLS) estimator is the best linear unbiased estimator for model parameters when the predictor variables are independent. The Multicollinearity problem arises when there exists near linear dependence among the predictor variables. This problem leads to variance inflation of the OLS estimator. Thus the interpretations based on it are not true. The ridge and Liu-type estimators are two methods to combat Multicollinearity. The Liu-type estimator is more efficient than the ridge estimator when there is a strong correlation between the predictor variables. We suppose that there is some prior information about parameter vector ,under a linear restriction as R,= r where R is a p2 ,p matrix and r is a p2 ,1 vector. The restricted estimator of ,is obtained by maximizing the log-likelihood function of the Linear regression model under the linear restriction. The Liu-type restricted estimator can be defined in the presence of Multicollinearity under the linear restriction. We propose the Liu-type Shrinkage estimators using the Liu-type and Liu-type restricted estimators to improve the estimation of parameters. We compare the performance of the Liu-type Shrinkage estimators and the Liu-type estimator in terms of their relative efficiency using a Monte Carlo simulation study. The simulation is conducted under different sample sizes, n = 30,50, the correlation level between the predictor variables ,= 0: 80,0: 90,0: 95, p1 = 5, and p2 = 3,5,7. To investigate the behavior of the proposed estimators, we define Δ,= ∥, ,􀀀, , 0∥, 2, where ∥, : ∥,is the Euclidean norm, ,is the parameters vector in the simulated model and , 0 is the true parameters vector in the candidate sub-model. We also apply the proposed estimation methods to a real data set. Results and Discussion The simulation results show that all estimators’,performances become better when p2 and ,increase for fixed n. For all combinations of p2, , , and n, the Liu-type restricted estimator has the best performance at Δ,= 0. As Δ,moves away from zero, all estimators’,simulated relative efficiencies (SREs) decrease. As ,approaches one, the performance of the Liu-type linear shrinkage estimator increases. Conclusion This paper suggested the Liu-type Shrinkage estimators in the Linear regression model in the presence of Multicollinearity under the subspace information. A Monte Carlo simulation was conducted to compare the proposed estimators’,performance with the Liu-type estimator. The simulation results confirm that the proposed estimators perform better than the Liu-type estimator when Δ,= 0 and near it for all p2, , , and n.

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APA: Copy

ZANDI, Z., & BEVRANI, H.. (2023). Coefficients Estimation of Linear Regression Models Using Liu-Type Shrinkage Estimators. JOURNAL OF STATISTICAL SCIENCES, 16(2 ), 417-434. SID. https://sid.ir/paper/1021862/en

Vancouver: Copy

ZANDI Z., BEVRANI H.. Coefficients Estimation of Linear Regression Models Using Liu-Type Shrinkage Estimators. JOURNAL OF STATISTICAL SCIENCES[Internet]. 2023;16(2 ):417-434. Available from: https://sid.ir/paper/1021862/en

IEEE: Copy

Z. ZANDI, and H. BEVRANI, “Coefficients Estimation of Linear Regression Models Using Liu-Type Shrinkage Estimators,” JOURNAL OF STATISTICAL SCIENCES, vol. 16, no. 2 , pp. 417–434, 2023, [Online]. Available: https://sid.ir/paper/1021862/en

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