This paper presents three lectures on a robust analysis of linear models. One of the main goals of these lectures is to show that this analysis, similar to the traditional least squares-based analysis, offers the user a unified methodology for inference procedures in general linear models. This discussion is facilitated throughout by the simple geometry underlying the analysis. The traditional analysis is based on the least squares fit which minimizes the Euclidean norm, while the robust analysis is based on a fit which minimizes another norm. Several examples involving real data sets are used in the lectures to help motivate the discussion.

Testing hypotheses about variance parameters arises in contexts where uniformity is important and also in relation to checking assumptions as a preliminary to analysis of variance (ANOVA), dose-response modeling, discriminant analysis and so forth. In contrast to procedures for tests on means, tests for variances derived assuming normality of the parent populations are highly nonrobust to nonnormality. Procedures that aim to achieve robustness follow three types of strategies: (1) adjusting a normal-theory test procedure using an estimate of kurtosis, (2) carrying out an ANOVA on a spread variable computed for each observation and (3) using resampling of residuals to determine p values for a given statistic. We review these three approaches, comparing properties of procedures both in terms of the theoretical basis and by presenting examples. Equality of variances is first considered in the two-sample problem followed by the k-sample problem (one-way design).

We consider the use of sign statistics in two different types of one-way layouts. The first layout is for data collected to compare several treatments. The second layout is for independent repeated measures on several subjects. In the first case we discuss hypothesis testing and multiple comparisons. In the second case we fit mixture models. We then show how fitting mixture models can be helpful in follow-up multiple comparisons in the first case.

This paper provides a practical description of density estimation based on kernel methods. An important aim is to encourage practicing statisticians to apply these methods to data. As such, reference is made to implementations of these methods in R, S-PLUS and SAS.

Multivariate nonparametric statistical tests of hypotheses are described for the one-sample location problem, the several-sample location problem and the problem of testing independence between pairs of vectors. These methods are based on affine-invariant spatial sign and spatial rank vectors. They provide affine-invariant multivariate generalizations of the univariate sign test, signed-rank test, Wilcoxon rank sum test, Kruskal–Wallis test, and the Kendall and Spearman correlation tests. While the emphasis is on tests of hypotheses, certain references to associated affine-equivariant estimators are included. Pitman asymptotic efficiencies demonstrate the excellent performance of these methods, particularly in heavy-tailed population settings. Moreover, these methods are easy to compute for data in common dimensions.

Quantile–quantile (QQ) plots for comparing two distributions are constructed by matching like-positioned values (i.e., quantiles) in the two distributions. These plots can reveal outliers, differences in location and scale, and other differences between the distributions. A particularly useful application is comparing residuals from an estimated linear model to the normal. A robust estimate, such as the Sen–Theil estimate, of the regression line is important. Extensions to two-dimensional QQ plots are presented, relying on a particular notion of multivariate position.

Some classes of nonparametric procedures with randomly right-censored data are presented. They include procedures for analysis of variance and analysis of covariance designs with independent and dependent ordinal (continuous and discrete) data.

When the distribution of X∈ℝ^p depends only on its distance to some θ₀∈ℝ^p, we discuss results from Hössjer and Croux and Neeman and Chang on rank score statistics. Similar results from Neeman and Chang are also given when X and θ₀ are constrained to lie on the sphere in ℝ^p. Results from Ko and Chang on M estimation for spatial models in Euclidean space and the sphere are also discussed. Finally we discuss a regression type model: the image registration problem. We have landmarks u_i on one image and corresponding landmarks V_i on a second image. It is desired to bring the two images into closest coincidence through a translation, rotation and scale change. The techniques and principles of this paper are summarized through extensive discussion of an example in three-dimensional image registration and a comparison of the L₁ and L₂ registrations. Two principles are important when working with spatial statistics: (1) Assumptions, such as that the distribution of X depends only on its distance to θ₀, introduce symmetries to spatial models which, if properly used, greatly simplify statistical calculations. These symmetries can be expressed in a more general setting by using the notion of statistical group models. (2) When working with a non-Euclidean parameter space Θ such as the sphere, techniques of elementary differential geometry can be used to minimize the distortions caused by using a coordinate system to reexpress Θ in Euclidean parameters.

This paper is intended to provide the reader with an introduction to ranked set sampling, a statistical technique for data collection that generally leads to more efficient estimators than competitors based on simple random samples. Methods for obtaining ranked set samples are described, and the structural differences between ranked set samples and simple random samples are discussed. Properties of the sample mean associated with a balanced ranked set sample are developed. A nonparametric ranked set sample estimator of the distribution function is discussed and properties of a ranked set sample analog of the Mann–Whitney–Wilcoxon statistic are presented.

Probabilistic and statistical models for the occurrence of a recurrent event over time are described. These models have applicability in the reliability, engineering, biomedical and other areas where a series of events occurs for an experimental unit as time progresses. Nonparametric inference methods, in particular, the estimation of a relevant distribution function, are described.

Quantile and conditional quantile statistical thinking, as I have innovated it in my research since 1976, is outlined in this comprehensive survey and introductory course in quantile data analysis. We propose that a unification of the theory and practice of statistical methods of data modeling may be possible by a quantile perspective. Our broad range of topics of univariate and bivariate probability and statistics are best summarized by the key words. Two fascinating practical examples are given that involve positive mean and negative median investment returns, and the relationship between radon concentration and cancer.

An introduction to nonparametric regression is accomplished with selected real data sets, statistical graphics and simulations from known functions. It is pedagogically effective for many to have some initial intuition about what the techniques are and why they work. Visual displays of small examples along with the plots of several types of smoothers are a good beginning. Some students benefit from a brief historical development of the topic, provided that they are familiar with other methodology, such as linear regression. Ultimately, one must engage the formulas for some of the linear curve estimators. These mathematical expressions for local smoothers are more easily understood after the student has seen a graph and a description of what the procedure is actually doing. In this article there are several such figures. These are mostly scatterplots of a single response against one predictor. Kernel smoothers have series expansions for bias and variance. The leading terms of those expansions yield approximate expressions for asymptotic mean squared error. In turn these provide one criterion for selection of the bandwidth. This choice of a smoothing parameter is done a rich variety of ways in practice. The final sections cover alternative approaches and extensions. The survey is supplemented with citations to some excellent books and articles. These provide the student with an entry into the literature, which is rapidly developing in traditional print media as well as on line.

The use of permutation methods for exact inference dates back to Fisher in 1935. Since then, the practicality of such methods has increased steadily with computing power. They can now easily be employed in many situations without concern for computing difficulties. We discuss the reasoning behind these methods and describe situations when they are exact and distribution-free. We illustrate their use in several examples.

Multivariate statistics plays a role of ever increasing importance in the modern era of information technology. Using the center-outward ranking induced by the notion of data depth, we describe several nonparametric tests of location and scale differences for multivariate distributions. The tests for location differences are derived from graphs in the so-called DD plots (depth vs. depth plots) and are implemented through the idea of permutation tests. The proposed test statistics are scale-standardized measures for the location difference and they can be carried out without estimating the scale or variance of the underlying distributions. The test for scale differences introduced in Liu and Singh (2003) is a natural multivariate rank test derived from the center-outward depth ranking and it extends the Wilcoxon rank-sum test to the testing of multivariate scale. We discuss the properties of these tests, and provide simulation results as well as a comparison study under normality. Finally, we apply the tests to compare airlines’ performances in the context of aviation safety evaluations.

The classical theory of rank-based inference is essentially limited to univariate linear models with independent observations. The objective of this paper is to illustrate some recent extensions of this theory to time-series problems (serially dependent observations) in a multivariate setting (multivariate observations) under very mild distributional assumptions (mainly, elliptical symmetry; for some of the testing problems treated below, even second-order moments are not required). After a brief presentation of the invariance principles that underlie the concepts of ranks to be considered, we concentrate on two examples of practical relevance: (1) the multivariate Durbin–Watson problem (testing against autocorrelated noise in a linear model context) and (2) the problem of testing the order of a vector autoregressive model, testing VAR(p₀) against VAR(p₀+1) dependence. These two testing procedures are the building blocks of classical autoregressive order-identification methods. Based either on pseudo-Mahalanobis (Tyler) or on hyperplane-based (Oja and Paindaveine) signs and ranks, three classes of test statistics are considered for each problem: (1) statistics of the sign-test type, (2) Spearman statistics and (3) van der Waerden (normal score) statistics. Simulations confirm theoretical results about the power of the proposed rank-based methods and establish their good robustness properties.

We illustrate nonparametric, and particularly rank-based analyses of crossover studies, designs in which each subject receives more than one treatment over time. Principles involved in using the Wilcoxon rank sum test in the simple two-period, two-treatment crossover are described through theory and example. We then extend the ideas to two-treatment designs with more than two periods and to three-treatment, three-period designs. When more than one nonparametric approach is available, we consider the issue of statistical power in choosing an appropriate test.

Herbert Aron David was born in Berlin, Germany, on December 19, 1925. He earned a Ph.D. (1953) in statistics from University College London. Since then he has been a Research Officer at CSIRO, Sydney (1953–1955), Senior Lecturer in Statistics at Melbourne University (1955–1957), Professor of Statistics at Virginia Polytechnic Institute (1957–1964), Professor of Biostatistics at the University of North Carolina (1964–1972), Director and Head, Statistical Laboratory and Department of Statistics at Iowa State University (1972–1984) and Distinguished Professor in Liberal Arts and Sciences at Iowa State University (1980–1996). Currently he is Emeritus Distinguished Professor in Liberal Arts and Sciences at Iowa State University. He is a Fellow of the American Statistical Association, the American Association for the Advancement of Science and the Institute of Mathematical Statistics. He is an elected member of the International Statistical Institute. He also has served as Editor of Biometrics and as President of the Biometric Society. A list of all his publications until 1996 can be found in Statistical Theory and Applications. Papers in Honor of Herbert A. David (1996) 313–326.

This interview was conducted on September 9, 2000 (Saturday) at the Department of Statistics, Iowa State University, Ames.