Statistical computing support for Lp estimation in augmented linear models under linear inequality restrictions

Abstract

<p>This research project deals with computationally related problems in the general area of l(,p) (p (GREATERTHEQ) 1) estimation in linear models. Methods for computing l(,p) estimate in linear models are studied. In case of p = 1, descent methods from Bloomfield and Steiger, and Usow are discussed. A proof of convergence of these methods is provided. In case of p > 1, Newton's method and Quasi-Newton method are discussed. A new method is proposed and studied. It performs extremely well for p close to 2. Also, closed form solutions of the l(,p) estimation problem having design matrix of dimension (m + 1) x m or (m +2) x m are derived, and methods of generating test problems for the general l(,p) estimation problem are discussed. In another part of the research project, the objective function for computing l(,p) estimate, augmented by the p('th) power of l(,p) norm of the parameter vector, has been studied. One result of this study is a way to identify the l(,p) estimate having the least l(,p) norm. Finally, branch-and-bound method for computing l(,p) estimate of linear models under linear inequality restrictions are discussed.</p>