Frank R. Hampel

The Influence Curve and its Role in Robust Estimation

Abstract This paper treats essentially the first derivative of an estimator viewed as functional and the ways in which it can be used to study local robustness properties. A theory of robust estimation “near” strict parametric models is briefly sketched and applied to some classical situations. Relations between von Mises functionals, the jackknife and U-statistics are indicated. A number of classical and new estimators are discussed, including trimmed and Winsorized means, Huber-estimators, and more generally maximum likelihood and M-estimators. Finally, a table with some numerical robustness properties is given.

The Breakdown Points of the Mean Combined With Some Rejection Rules

In the past, methods for rejection of outliers have been investigated mostly without regard to the quantitative consequences for subsequent estimation or testing procedures. Moreover, although rejection of outliers with subsequent application of least squares methods is one of the oldest and most widespread classes of robust procedures, until recently no comparison was made with other robust methods. In this article the simplest situation, namely estimation of a location parameter in the potential presence of outliers, is treated by means of a Monte Carlo study. This study yields Monte Carlo variances of the “arithmetic mean” after rejection of outliers according to several classical and recent formal rules. The results are also compared with those for other robust estimators of location parameters. It turns out that a simple summary and theoretical explanation of the Monte Carlo results is provided by the breakdown points of the combined rejection-estimation procedures. As a by-product, the concept of br...

The Change-of-Variance Curve and Optimal Redescending M-Estimators

Abstract We define the change-of-variance curve (CVC) of location M-estimators in order to investigate the infinitesimal stability of the asymptotic variance. We also construct the so-called hyperbolic tangent estimators, proving their existence and performing certain numerical computations of their defining constants. Their introduction is motivated by a theorem that shows they are the optimally robust redescending M-estimators in the sense of the CVC.

Is statistics too difficult

By means of several historical examples, it is shown that it does not appear to be easy to build bridges between rigorous mathematics and reasonable data-analytic procedures for scientific measurements. After mentioning both some positive and some negative aspects of statistics, a formal framework for statistics is presented which contains the concept formation, derivation of results and interpretation of mathematical statistics as three essential steps. The difficulties especially of interpretation are shown for examples in several areas of statistics, such as asymptotics and robustness. Some problems of statistics in two subject-matter sciences are discussed, and a summary and outlook are given.

Modern trends in the theory of robustness 2

The first part of this paper contains an introduction into robust statistics with special emphasis on some difficult conceptual points and on recent results concerning the intuitive basis of robustness theory. A brief review of some classical robustness concepts and results follows, and the last part discusses a number of research areas in robustness which are of current interest, including treatment of arbitrary unstructured data and of unbalanced linear models.

Nonadditive Probabilities in Statistics

The paper gives a short survey about the occurrence (sometimes hidden in the background) of nonadditive probabilities in statistics. It starts with the original meaning of “probability” in statistics in the Ars Conjectandi by Jakob (James) Bernoulli, and the ensuing misunderstanding which gave the term its present meaning. One chapter is about robustness theory, its use of (nonadditive) Choquet-capacities, and an attempt to clarify some widespread misunderstandings about it, which have consequences for the use of upper and lower probabilities. Also the uncertainty about model choice (including the conflict between purely mathematical reasoning and good statistical practice) and treatment of outliers is briefly discussed.The partial arbitrariness of additivity both in Bayes’ famous Scholium and in modern Bayes theory is outlined. The infamous and almost forgotten fiducial probabilities can actually be corrected and find their place in a more general paradigm using upper and lower probabilities. Finally, a new (?) qualitative theory of inference is mentioned which may contain some essentials of inductive reasoning in real life.

A smoothing principle for the Huber and other location M-estimators

A smoothing principle for M-estimators is proposed. The smoothing depends on the sample size so that the resulting smoothed M-estimator coincides with the initial M-estimator when n->~. The smoothing principle is motivated by an analysis of the requirements in the proof of the Cramer-Rao bound. The principle can be applied to every M-estimator. A simulation study is carried out where smoothed Huber, ML-, and Bisquare M-estimators are compared with their non-smoothed counterparts and with Pitman estimators on data generated from several distributions with and without estimated scale. This leads to encouraging results for the smoothed estimators, and particularly the smoothed Huber estimator, as they improve upon the initial M-estimators particularly in the tail areas of the distributions of the estimators. The results are backed up by small sample asymptotics.

On the Philosophical Foundations of Statistics: Bridges to Huber’s work, and recent results

After a few words in honor of P.J. Huber and a brief introduction into a new frequentist paradigm for the foundations of statistics, which uses upper and lower probabilities and the concept of “successful bets”, the talk discusses some very recent research results with more general implications: Successful bets for the normal and the exponential shift model, asymptotically successful bets, and a sketch for approximately successful bets in the context of robust statistics.

What can the foundations discussion contribute to data analysis? And what may be some of the future directions in robust methods and data analysis?

Abstract The discussion of the philosophical foundations of statistics is not necessarily far remote from practical data analysis, even though, fortunately, a good data analysis does not depend very much on the philosophical framework adopted. Some clarification for data analysis can be provided about the purposes of estimation and testing, the choice between parameter estimation and prediction, the role of parametric models, the difference between epistemic and physical probabilities, the interpretation of confidence intervals and fiducial probabilities, the description of a state of partial knowledge, including the use of upper and lower probabilites, and other topics. This paper contains first some remarks on future directions in robust methods, in loose connection with foundations, but in close connection with the title of the workshop. It then gives a brief description of a new approach to the foundations of statistics, and in the main part it discusses a number of suggestions and consequences for data analysis as mentioned above, which can be derived from the study and discussion of the foundations of statistics.

The proper fiducial argument

The paper describes the proper interpretation of the fiducial argument, as given by Fisher in (only) his first papers on the subject. It argues that far from being a quaint, little, isolated idea, this was the first attempt to build a bridge between aleatory probabilities (the only ones used by Neyman) and epistemic probabilities (the only ones used by Bayesians), by implicitly introducing, as a new type, frequentist epistemic probabilities. Some (partly rather unknown) reactions by other statisticians are discussed, and some rudiments of a new, unifying general theory of statistics are given which uses upper and lower probabilities and puts fiducial probability into a larger framework. Then Fishers pertaining 1930 paper is being reread in the light of present understanding, followed by some short sections on the (legitimate) aposteriori interpretation of confidence intervals, and on fiducial probabilities as limits of lower probabilities.