| Least squares regression is far from optimal in many non-Gaussian situations, especially when the errors follow distributions with longer tails or when the data occur with outlying observations.;Least absolute value (LAV) regression overcomes these drawbacks and provides an attractive alternative to least squares regression. LAV regression is less sensitive than least squares to extreme errors, and it also yields maximum likelihood estimators if the errors follow the Laplace distribution (double-exponential distribution).;In LAV regression the problem is to find the value of the parameters in the linear model so as to minimize the absolute value of the sum of the error terms. The problem has been formulated as a linear programming problem. Procedures have only recently been developed for testing hypotheses about the parameters of the model. Probably the principal difficulty with these statistical procedures is that the theory requires an estimate of the density function evaluated at the median of a sample from the error distribution.;Under certain circumstances a stepwise regression procedure may be desirable, as opposed to other methods of variable selection, especially when a large number of variables is involved. There are a number of computer programs that perform stepwise regression using the least squares criterion.;In this work we develop an algorithm for adding and deleting variables from the regression equation to identify finally a single subset of variables that is "good" according to a prespecified criterion for predictive purposes. To economize on computation each time we wish to add or to delete a variable, we start with the optimal solution of the previous problem instead of solving the current problem outright. The starting problem includes only the constant term of the regression equation.;In order to determine when the addition or deletion of a variable gives a statistically significant effect, we make use of the previously developed large sample theory to estimate the variance of the sample median. This theory requires an estimate of the density function for the error terms evaluated at its median. We investigate a number of estimators, both new and old, and conclude that the least squares quadratic estimator of the density function, a new estimator, is clearly superior to the others. Finally, we write a computer program that performs an LAV stepwise regression. (Abstract shortened with permission of author.). |
