James E. Gentle

Least absolute values estimation: an introduction

A brief review and bibliography of least absolute values (LAV) estimation is given. This paper serves to introduce the other articles in this special issue on the computational aspects of LAV estimation.

On least absolute values estimation

The resistance of least absolute values (L1) estimators to outliers and their robustness to heavy-tailed distributions make these estimators useful alternatives to the usual least squares estimators. The recent development of efficient algorithms for L1 estimation in linear models has permitted their use in practical data analysis. Although in general the L1 estimators are not unique, there are a number of properties they all share. The set of all L1 estimators for a given model and data set can be characterized as the convex hull of some extreme estimators. Properties of the extreme estimators and of the L1-estimate set are considered.

A robust test for the homogeneity of scales

Many robust tests for the equality of variances have been proposed recently. Brown and Forsythe (1974) and Layard (1973) review some of the well-known procedures and compare them by simulation methods. Brown and Forsythe’s alternative formulation of Levene’s test statistic is found to be quite robust under certain nonnormal distributions. The performance of the methods, however, suffers in the presence of heavy tailed distributions such as the Cauchy distribution. In this paper, we propose and study a simple robust test. The results obtained from the Monte Carlo study compare favorably with those of the existing procedures.

A computer oriented method for generating test problems for L1 regression

A numerical method for obtaining data (X|y), relative to the linear model , is given. The user is allowed to specify column means for the X matrix, the general order of condition number, the unique L1 solution vector and the deviations of ys about the fitted hyperplane. Implementation of the method requires little more than use of subroutines found in most modern subroutine libraries. Computer generated data of this kind are useful in numerical studies of the operating characteristics of different algorithms.

Useful generalized properties of L1 estimators

Recent results by G. Appa and C. Smith, as well as I. Barrodale and F. D. K. Roberts, underscore several properties exhibited for fitting a linear model to a set of observation points under the criterion of least sum of absolute deviations(commonly denoted as the L1 criterion). This paper will generalize these properties to the non-full rank case and relax in a natural way some assumptions given by Appa and Smith.

Testing for Outliers in Linear Regression

The problem of outliers in the regression model is considered. For the case of one outlier at most, the use of the maximum absolute studentized residual, Rn, for identification of the outlier has been suggested by a number of authors. Simulation studies of the power of a conservative test based on Rn for identifying single outliers in regression models with one, two, and three independent variables are reported. The case of multiple outliers is also considered and techniques for their identification are discussed. A simulation study of a sequential procedure for handling two outliers is reported.

On the Distribution of the Studentized Bivariate Range

The stltdetltized bivnhte range, R s , it1 a sample from a circrdur normal distribution is espressed as the maximum of a set of F variables. Using a method employed by pearson and Chandra Sekar [7], the exact upper tail distribution for small samples is given. The same procedure provides over-approximations to the percentage points of R s , for larger sample sizes. Methods for obtaining the full distribution are discrussed. The use of the statistic in testitrg for homogeneity is considered.

A robust test for the coincidence of regressions

When the disturbances in regression models do not have normal distributions, the usual F-test for coincidence of regressions is not appropriate. We propose a simple robust test for coincidence of regressions and by means of a simulation study compare its performance with that of the usual F-test for models with errors having distributions from the family of symmetric stable distributions.

On the asymptotic bias of grenander’s mode estimator

Grenander introduced a direct estimator of the mode for a large class of densities. This note considers a large subclass of these densities for which Grenander’s estimator is asymptotically biased. Some of the distributions from this subclass include the F, gamma, and beta for which asymptotic expressions for the bias are given. To reduce the bias, it is recommended to choose larger values for one of the parameters of the estimator when the underlying distribution is nonsymmetric.

Examining rounding error in least absolute values regression computations

Two techniques for detecting inaccuracies in least absolute values (LAV) regression computations are presented and discussed. Examples of the use of the methods are given. The techniques are shown to apply to the more general case of M-estimation.