A Proposed Nth – Order Jackknife Ridge Estimator for Linear Regression Designs

Abstract

Several remediation measures have been developed to circumvent the problem of collinearity in General Linear Regression Designs. These include the Generalized Ridge, Jackknife Ridge, second- order Jackknife Ridge estimation procedures. In this paper, an nth-order Jackknife Ridge estimator is developed using canonical parameter transformation. Using the MATLAB version 7 software, parameter estimates, biases and variances of these estimators are computed to show their behavior and strengths. The results show that the parameter estimates are basically the same for all the methods. There is variance reduction at the Generalized Ridge estimator and at the ordered Jackknife Ridge estimators, though the Generalized Ridge estimator is slightly superior in this respect. As the order of Jackknife Ridge estimator increases, the variance decreases up to a certain nth-order and remains constant thereafter. Where variances of two consecutive estimators are the same or nearly so, the last but one estimator is considered optimal. This establishes a convergence criterion for the sequence of Jackknife Ridge estimators. It is shown from the five illustrative design matrices that higher order Jackknife Ridge estimators are superior to lower order Jackknife Ridge estimators in terms of bias. Thus further solving the problem of bias introduced by the Ridge estimator.

Keywords: Canonical transformation; collinearity; mean square error; positive definite matrix; squared bias; variance inflation.