Performance of Some New Ridge Parameters in Two-Parameter Ridge Regression Model

Abstract

Two-parameter ridge regression is a widely used method in the last two decades to circumvent the problem of multicollinearity. Ridge parameter k plays an important role in such situations. Several methods are available in the literature for the estimation of ridge parameter. For high multicollinearity, the available methods do not perform well in terms of mean square error. In this article, we propose some new estimators for the ridge parameter. Based on simulation study, our new estimators generally perform better than ordinary least squares estimator, ridge regression estimator and two-parameter ridge regression estimator in many considered scenarios especially for high multicollinearity. In addition, the new estimators also perform well for some non-normal error distributions. Finally, two real-life examples are used to illustrate the application of the proposed estimator.

A ridge estimation-sequence quadratic programming constraint optimization algorithm for data processing in multi-spectral thermometry

“…Then, the diagonal element r ij of matrix is the variance expansion factor of ridge estimation. According to experience, when k makes all r ij ⩽ 10, the ridge estimation will be relatively stable [22]. In a stable case, the error of ridge estimation is often greater than that of least squares estimation [23].…”

Weighted ridge and Liu estimators for linear regression model

“…Some contemporary research also presented on two parameter estimator to cope with the problem of multicollinearity. These authors include References 5–22. Liu 13 identified that in case of severe multicollinearity, the RRE and the LE may not fully address the ill‐conditioned problem and suggested a two parameter LE by extending Equation () as $$$$ -{k}^{-1/2}d{\hat{\beta}}_{OLS}=\beta +\varepsilon, \kern0.6em -\infty \le d\le \infty, \kern0.36em k>0 $$$$ and the following two parameter LE is obtained: $$$$ \hat{\beta}\left(k,d\right)={\left({X}^{\prime }X+ kI\right)}^{-1}\left({X}^{\prime }y-d{\beta}^{\ast}\right),\kern0.36em k>0,&#x002D;&#x0005C;infty <d<\infty, $$$$ where k is the ridge parameter that control the condition number of $$$ {X}^{\prime }X+ kI $$$ to the desired level and d is the Liu parameter that is used to improve the fit and $$$ {\beta}^{\ast } $$$ can be any estimator of $$$ \beta $$$.…”