Liu Estimator Based on An M Estimator

Objective: In multiple linear regression analysis, multicollinearity and outliers are two main problems. In the presence of multicollinearity, biased estimation methods like ridge regression, Stein estimator, principal component regression and Liu estimator are used. On the other hand, when outliers exist in the data, the use of robust estimators reducing the effect of outliers is prefered. Material and Methods: In this study, to cope with this combined problem of multicollinearity and outliers, it is studied Liu estimator based on M estimator (Liu M estimator). In addition, mean square error (MSE) criterion has been used to compare Liu M estimator with Liu estimator based on ordinary least squares (OLS) estimator. Results: OLS, Huber M, Liu and Liu M estimates and MSEs of these estimates have been calculated for a data set which has been taken form a study of determinants of physical fitness. Liu M estimator has given the best performance in the data set. It is found as both MSE (?LM) = 0.0078< MSE (?M) = 0.0508 and MSE (?LM) = 0.0078< MSE (?L)= 0.0085. Conclusion: When there is both outliers and multicollinearity in a dataset, while using of robust estimators reduces the effect of outliers, it could not solve problem of multicollinearity. On the other hand, using of biased methods could solve the problem of multicollinearity, but there is still the effect of outliers on the estimates. In the occurence of both multicollinearity and outliers in a dataset, it has been shown that combining of the methods designed to deal with this problems is better than using them individually.