Lp norm estimation procedures and an L1 norm algorithm for unconstrained and constrained estimation for linear models

When the distribution of the errors in a linear regression model departs from normality, the method of least squares seems to yield relatively poor estimates of the coefficients. One alternative approach to least squares which has received a great deal of attention of late is minimum L_p norm estimation. However, the statistical efüciency of a L_p estimator depends greatly on the underlying distribution of errors and on the value of p. Thus, the choice of an appropriate value of p is crucial to the effectiveness of _p estimation. Previous work has shown that L₁ estimation is a robust procedure in the sense that it leads to an estimator which has greater statistical efficiency than the least squares estimator in the presence of outliers, and that L₁ estimators have some- desirable statistical properties asymptotically. This dissertation is mainly concerned with the development of a new algorithm for L₁ estimation and constrained L₁ estimation. The mainstream of computational procedures for L₁ estimation has been the simplex-type algorithms via the linear programming formulation. Other procedures are the reweighted least squares method, and. nonlinear programming technique using the penalty function approach or descent method. A new computational algorithm is proposed which combines the reweighted least squares method and the linear programming approach. We employ a modified Karmarkar algorithm to solve the linear programming problem instead of the simplex method. We prove that the proposed algorithm converges in a finite number of iterations. From our simulation study we demonstrate that our algorithm requires fewer iterations to solve standard problems than are required by the simplex-type methods although the amount of computation per iteration is greater for the proposed algorithm. The proposed algorithm for unconstrained L₁ estimation is extended to the case where the L₁ estimates of the parameters of a linear model satisfy certain linear equality and/or inequality constraints. These two procedures are computationally simple to implement since a weighted least squares scheme is adopted at each iteration. Our results indicate that the proposed L₁ estimation procedure yields very accurate and stable estimates and is efficient even when the problem size is large.

Name thesis:degree_name

Ph. D.

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Statistics

Department dc:contributor.department

Statistics

Grantor dc:publisher

Virginia Polytechnic Institute and State University

Year dc:date.issued

1986

Author dc:creator

• Kim, Buyong

Chair dc:contributor.committeechair

• Skarpness, Bradley O.

Committee members dc:contributor.committeemember

• Foutz, Robert
• Reynolds, Marion R. Jr.
• Krutchkoff, Richard G.
• Sherali, Hanif D.

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Language dc:language.iso

en_US

Handle dc:identifier.uri

http://hdl.handle.net/10919/53627

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/53627