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## 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<sub>p</sub> norm estimation. However, the statistical efüciency of a L<sub>p</sub> 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 <sub>p</sub> 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. / Ph. D.

Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/53627 |

Date | January 1986 |

Creators | Kim, Buyong |

Contributors | Statistics, Skarpness, Bradley O., Foutz, Robert, Reynolds, Marion R. Jr., Krutchkoff, Richard G., Sherali, Hanif D. |

Publisher | Virginia Polytechnic Institute and State University |

Source Sets | Virginia Tech Theses and Dissertation |

Language | en_US |

Detected Language | English |

Type | Dissertation, Text |

Format | ix, 133 leaves, application/pdf, application/pdf |

Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |

Relation | OCLC# 15280160 |

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