## Retrospective Theses and Dissertations

#### Title

Statistical aspects of L1 regression

Dissertation

1985

#### Degree Name

Doctor of Philosophy

Statistics

#### Abstract

Consider the linear model Y = X(beta) + (epsilon), where Y is an n x 1 vector of response variables; X is an n x p matrix of values of concomitant variables with entries x(,i,j), i = 1,2, ..., n; j = 0,1, ..., p-1; (beta) is a p x 1 vector of parameters ((beta)(,0), (beta)(,1), ..., (beta)(,p-1)); and (epsilon) is an n x 1 vector of random disturbances, assumed to be independent and identically distributed, each following some distribution function F;Historically, the residuals ((epsilon)) have been assumed to be indepen- dent and identically distributed normal random variables with mean zero and variance (sigma)('2). This assumption has led to almost exclusive use of the least squares criterion for estimating (beta). However, the assumption of normally distributed residuals is currently being questioned for many applications; when the normality of residuals is in doubt, so is use of the least squares regression criterion. There are many possible alternatives to least squares, and this dissertation addresses one of these: the L(,1) regression criterion, also frequently referred to as minimum absolute deviations (MAD) or least absolute;values (LAV) regression. L(,1) regression consists of finding estimates (')(beta) of (beta) such that;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);Techniques for obtaining L(,1) estimates have been the subject of a great deal of research and are only summarized in this dissertation. However, there has not been much done to develop inference tech- niques for L(,1) regression; this dissertation tries to develop such techniques. The inferences developed here are for (beta)(,1), the slope of the simple regression line (the linear model given above, with p = 2). Three different strategies are employed to provide prospective inferences: in Chapter 2, nonparametric techniques arising from L(,1) estimation are explored; in Chapter 3, the asymptotic distribution of (')(beta)(,1), the L(,1) estimator of (beta)(,1), is used to generate inferences; in Chapter 4, likelihood ratio statistics are derived assuming residuals are distributed double exponential, and inferences based on these statistics are developed.

#### DOI

https://doi.org/10.31274/rtd-180813-12747

#### Publisher

Digital Repository @ Iowa State University, http://lib.dr.iastate.edu/

Michael David Tveite

en

AAI8604523

application/pdf

135 pages

COinS