Ridge Estimator (RE) has been used as an alternative estimator for Ordinary Least Squared Estimator (OLSE) to handle multicollinearity problem in the linear regression model. However, it introduces heavy bias when the number of predictors is high, and it may shrink irrelevant regression coefficients, but they are still in the model. Least Absolute Shrinkage and Selection Operator (LASSO) and Elastic net (Enet) estimator have been used to make the variable selection and shrinking the regression coefficients simultaneously. Further, the model misspecification due to excluding relevant explanatory variable in the linear regression model is considered as a severe problem in statistical research, and it will lead to bias and inconsistent parameter estimation. The performance of RE, LASSO and Enet estimators under the correctly specified regression model was well studied in the literature. This study intends to compare the performance of RE, LASSO and Enet estimators in the misspecified regression model using Root Mean Square Error (RMSE) criterion. A Monte-Carlo simulation study was used to study the performance of the estimators. In addition to that, a real-world example was employed to support the results. The analysis revealed that Enet outperformed RE and LASSO in both correctly specified model and misspecified regression model.
In this paper, the performance of existing biased estimators (Ridge Estimator (RE), Almost Unbiased Ridge Estimator (AURE), Liu Estimator (LE), Almost Unbiased Liu Estimator (AULE), Principal Component Regression Estimator (PCRE), r-k class estimator and r-d class estimator) and the respective predictors were considered in a misspecified linear regression model when there exists multicollinearity among explanatory variables. A generalized form was used to compare these estimators and predictors in the mean square error sense. Further, theoretical findings were established using mean square error matrix and scalar mean square error. Finally, a numerical example and a Monte Carlo simulation study were done to illustrate the theoretical findings. The simulation study revealed that LE and RE outperform the other estimators when weak multicollinearity exists, and RE, r-k class and r-d class estimators outperform the other estimators when moderated and high multicollinearity exist for certain values of shrinkage parameters, respectively. The predictors based on the LE and RE are always superior to the other predictors for certain values of shrinkage parameters.
Among several variable selection methods, LASSO is the most desirable estimation procedure for handling regularization and variable selection simultaneously in the high-dimensional linear regression models when multicollinearity exists among the predictor variables. Since LASSO is unstable under high multicollinearity, the elastic-net (Enet) estimator has been used to overcome this issue. According to the literature, the estimation of regression parameters can be improved by adding prior information about regression coefficients to the model, which is available in the form of exact or stochastic linear restrictions. In this article, we proposed a stochastic restricted LASSO-type estimator (SRLASSO) by incorporating stochastic linear restrictions. Furthermore, we compared the performance of SRLASSO with LASSO and Enet in root mean square error (RMSE) criterion and mean absolute prediction error (MAPE) criterion based on a Monte Carlo simulation study. Finally, a real-world example was used to demonstrate the performance of SRLASSO.
The analysis of misspecification was extended to the recently introduced stochastic restricted biased estimators when multicollinearity exists among the explanatory variables. The Stochastic Restricted Ridge Estimator (SRRE), Stochastic Restricted Almost Unbiased Ridge Estimator (SRAURE), Stochastic Restricted Liu Estimator (SRLE), Stochastic Restricted Almost Unbiased Liu Estimator (SRAULE), Stochastic Restricted Principal Component Regression Estimator (SRPCRE), Stochastic Restricted r-k (SRrk) class estimator, and Stochastic Restricted r-d (SRrd) class estimator were examined in the misspecified regression model due to missing relevant explanatory variables when incomplete prior information of the regression coefficients is available. Further, the superiority conditions between estimators and their respective predictors were obtained in the mean square error matrix (MSEM) sense. Finally, a numerical example and a Monte Carlo simulation study were used to illustrate the theoretical findings.
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