Communications in Statistics: Theory and Methods

Hogg, Robert V.

doi: 10.1080/03610927708827531pmid: N/A

This introduction is given to make it easier for the statistician who knows little about robust methods to read the articles in this special issue. Various terms will be defined here, and there will be some explanation of how each of the individual articles relates to the total picture that we wish to present.

Wegman, Edward J.; Carroll, Raymond J.

doi: 10.1080/03610927708827532pmid: N/A

Andrews et al (1972) carried out an extensive Monte Carlo study of robust estimators of location. Their conclusions were that the hampel and the skipped estimates, as classes, seemed to be preferable to some of the other currently fashionable estimators. The present study extends this work to include estimators not previously examined. The estimators are compared over short-tailed as well as long-tailed alternatives and also over some dependent data generated by first-order autoregressive schemes. The conclusions of the present study are threefold. First, from our limited study, none of the so-called robust estimators are very efficient over short-tailed situations. More work seems to be necessary in this situation. Second, none of the estimators perform very well in dependent data situations, particularly when the correlation is large and positive. This seems to be a rather pressing problem. Finally, for long-tailed alternatives, the hampel estimators and Hogg-type adaptive versions of the hampels are the strongest classes. The adaptive hampels neither uniformly outperform nor are they outperformed by the hampels. However, the superiority in terms of maximum relative efficiency goes to the adaptive hampels. That is, the adaptive hampels, under their worst performance.

Holland, Paul W.; Welsch, Roy E.

doi: 10.1080/03610927708827533pmid: N/A

The rapid development of the theory of robust estimation (Huber, 1973) has created a need for computational procedures to produce robust estimates. We will review a number of different computational approaches for robust linear regression but focus on one—iteratively reweighted least-squares (IRLS). The weight functions that we discuss are a part of a semi-portable subroutine library called ROSEPACK (RObust Statistical Estimation PACKage) that has been developed by the authors and Virginia Klema at the Computer Research Center of the National Bureau of Economic Research, Inc. in Cambridge, Mass. with the support of the National Science Foundation. This library (Klema, 1976) makes it relatively simple to implement an IRLS regression package.

Leon Harter, H.

doi: 10.1080/03610927708827534pmid: N/A

The question of nonuniqueness of the least absolute values (L1) regression is discussed, and examples are given of situations where the L1-regression is unique and where it has 2, 3 and 4 limiting positions. A method is proposed for finding a compromise regression line when the L1 regression is not unique. Suggestions are made for further research.

Gentle, J. E.; Kennedy, W. J.; Sposito, V. A.

doi: 10.1080/03610927708827535pmid: N/A

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.

Lenth, Russell V.

doi: 10.1080/03610927708827536pmid: N/A

We consider the problem of fitting a cubic spline to data using robust regression techniques. Some important properties of splines are discussed, showing that their use as a regression model is related in principle to the concept of robustness. Methods for fitting splines and interpreting the results are outlined, and an illustrative example is given.

Hettmansperger, Thomas P.; Utts, Jessica M.

doi: 10.1080/03610927708827537pmid: N/A

Robustness properties of a family of rank estimates to compete with trimmed means and other robust estimates for the one sample location problem are investigated. In particular, the influence curve and breakdown point are developed, as well as their finite sample equivalents, the sensitivity curve and tolerance. The estimates are formulated from a one sample rank test rather than the customary two sample rank test approach. In addition, a functional is implicitly defined for the asymptotic version of the estimate. Computational problems are considered and a simple iterative procedure for finding the estimate is given.

Jones, Douglas H.

doi: 10.1080/03610927708827538pmid: N/A

An asymptotically efficient combination of rank statistics with a variety of score functions is studied. The results of a Monte Carlo study show that the power function of the test for location is very stable across density types varying from heavy to light tails.

Schrader, Ronald M.; Mc Kean, Joseph W.

doi: 10.1080/03610927708827539pmid: N/A

The classes of R- and M-estimates contain practical robust alternatives to least squares estimation in linear models. These estimates form the basis for a robust analysis of variance. This inference procedure is described and its versatility demonstrated.