Robert V. Hogg

Adaptive Robust Procedures: A Partial Review and Some Suggestions for Future Applications and Theory

Abstract After providing some background for the need to consider estimates other than those resulting from normal theory, there is a brief review of some nonadaptive robust estimators. We introduce adaptive estimators using those of Tukey and McLaughlin, Jaeckel, Johns, Birnbaum and Mike, Takeuchi, Hajek, van Eeden, and Beran. Adaptive estimators based on preliminary testing and Stein-like procedures are then considered, and recommendations are made on how to select the amount of trimming. Various proposals for estimating regression coefficients are also considered. Adaptive distrubution-free tests look very promising for improving the power of nonparametric tests, and some of these techniques can be used effectively in data analysis. Asymmetric trimmed means, adapted to the particular sample, can easily be used with data and provide good descriptive statistics having an approximate error structure. Finally, it is conjectured that estimators based on “cliff-hangers” might be extremely effective if there ...

Statistical Education: Improvements are Badly Needed

Thirty-nine statisticians gathered for a workshop on statistical education in Iowa City, IA, June 18-20, 1990. The workshop was sponsored by the University of Iowa and the American Statistical Association (ASA) with financial assistance provided by the National Science Foundation, the ALCOA Foundation, the Ott Foundation, the Statistics Division of the American Society for Quality Control, and the Quality and Productivity Section of the ASA. Many of the main suggestions resulting from this workshop were listed on pp. 19-20 in the November 1990 issue of Arnstat News. I attempt here to capture some of our thoughts on statistical education. I have been greatly influenced by the discussions at the workshop, which concentrated mainly on a first course, and thus I thank those in attendance. However, I do not want to imply that everyone there would agree with my remarks, and I hope many of them will write papers expressing their views. In particular, I note that the theme of the 1992 Winter Meeting of the ASA is statistical education, and many ideas on the subject will be presented there. There is a general feeling that students are not as well prepared for college-level science and mathematics courses as they were 20 to 30 years ago. Of course, there are many reasons for this, and I mention only a few. Parents can be blamed, at least partially; that is, academic achievement in the eyes of the public is not encouraged and respected enough today. Of course, school teachers must accept some of the responsibility too, although other considerations do create difficult situations for them. Moreover, students spend far too much time on other things, in particular, athletics and working long hours at

A Two-Sample Adaptive Distribution-Free Test

Can-hour jobs to make money to buy gas, stereos, junk food, and such. Unfortunately, it leaves many of them with little time for their most important activity: education. So we, as college teachers, recognize these problems, some of which are being addressed with new programs like the ASA’s Quantitative Literacy project. However, statisticians do have an obligation to try to do much bet-

An Introduction to Robust Estimation

Abstract An adaptive distribution-free test is proposed for the two-sample location problem. First, the data are used to assess the tailweight and skewness of the underlying distributions. This leads to the selection and then application, with the same data, of one of several common rank tests for shift, such as the Mann-Whitney-Wilcoxon test. The preliminary selection is made in a way that insures the testing procedure is distribution-free. A Monte Carlo study shows that the adaptive test has excellent power over a wide class of distributions and is preferable to certain prominent nonadaptive tests.

Some Observations on Robust Estimation

Publisher Summary This chapter provides an overview of robust estimation. It is recognized that outliers, which arise from heavy tailed distributions or are simply bad data points because of errors, have an unusually large influence on the least squares estimators. That is, the outliers pull the least squares fit toward them too much; a resulting examination of the residuals is misleading because then they look more like normal ones. Accordingly, robust methods have been created to modify least squares schemes so that the outliers have much less influence on the final estimates. One of the most satisfying robust procedures is that given by a modification of the principle of maximum likelihood. Robust methods have consequently been used successfully in many applications. There has been some evidence that adaptive procedures are of value. The basic idea of adapting is the selection of the estimation procedure after observing the data.

Statistical Robustness: One View of its use in Applications Today

Abstract Let Tj be a reasonable estimator (for example, a minimum mean square error estimator) of the parameter θ of the family Dj of distributions, j = 1, 2, …, m. An estimator T, which is a weighted mean of T 1, T s, …, Tm , is found that has the same asymptotic distribution as that of Tj , when the sample comes from Dj , j = 1, 2, …, m. Here the weights are functions of the sample items. Empirical evidence is given which indicates that T is satisfactory for small sample sizes. It is proved that if Tj and the weight Wj are odd location and even location-free statistics, respectively, j = 1, 2, …, m, then T = ΣWiTi , where ΣWi = 1, is an unbiased estimator of the center of every symmetric distribution, provided certain expectations exist. This is useful in the construction of the weight function Wj.

More Light on the Kurtosis and Related Statistics

Abstract Users of statistical packages need to be aware of the influence that outlying data points can have on their statistical analyses. Robust procedures provide formal methods to spot these outliers and reduce their influence. Although a few robust procedures are mentioned in this article, one is emphasized; it is motivated by maximum likelihood estimation to make it seem more natural. Use of this procedure in regression problems is considered in some detail, and an approximate error structure is stated for the robust estimates of the regression coefficients. A few examples are given. A suggestion of how these techniques should be implemented in practice is included.

On Constructing Statistics and Reporting Data

Abstract We develop a scale invariant test for one member of the family given by the symmetric densities c exp [—|x|θ], θ>0 against another member. The statistic which arises in testing θ = 2 (normal) against θ = 4 is then modified to make it also location invariant; this modification results in the kurtosis K of the sample. Three other statistics are proposed for testing light-tailed distributions against heavy-tailed ones; in particular, the ratio V of one-half of the range and the mean deviation from the sample median is considered. A Monte Carlo study compares the relative values of the four statistics.

Generalized Linear and Quadratic Discriminant Functions Using Robust Estimates

This article has two purposes: (a) To construct useful statistics by estimating important probabilities, a method that could easily be used in a first course in mathematical statistics as an alternative to standard methods (like the likelihood ratio). (b) To discuss these probability estimates as another method for reporting the findings of a quantitative stujdy. Since this idea has been used in certain selected nonparametric problems, it is not new. Nevertheless, its generality has not been explored, and its usefulness in reporting data has been seemingly overlooked. The basic idea is to consider one or more important probabilities that are associated with the problem under consideratiori and then to estimate these probabilities, assuming the most reasonable mathematical model available. We believe that these estimates, along with appropriate error bounds, will be as useful to the consumers of statistical analyses as are our present techniques. As a matter of fact, it would be easy, in a first course in mathematical statistics, to use this approach, as it produces nearly all of the standard parametric and nonparametric techniques associated with one or two distributions. Since the main purpose of our paper is to advertise this approach, we give several illustrations which demonstrate its adaptability, without going into any one in great depth. It is hoped that many statisticians, including the authors, will want to explore its various characteristics further.

Adaptive distribution-free tests

Abstract Two new methods of constructing robust linear and quadratic discriminant functions are introduced. The first is a generalization of Fishers procedure for finding a linear discriminant function. It places less weight on those observations that are far from the overlapping regions of the two populations. The second new method substitutes M-estimates of the means and the covariance matrices into the usual expressions for the linear and quadratic discriminant functions. Monte Carlo results indicate lower misclassification probabilities for these schemes compared to Fishers linear discriminant function in cases of heavy-tailed or contaminated distributions.