INTRODUCTION

Estimation is the art of inferring information about some unknown quantity on the basis of available data. Typically an estimator of some sort is used. The estimator is chosen to perform well under the conditions that are assumed to underly the data. Since these conditions are never known exactly, estimators must be chosen which are robust, which perform well under a variety of underlying conditions.

The theory of robust estimation is based on the specified properties of specified estimators under specified conditions. This book is the result of a large study designed to investigate the interaction of these three components over as wide a range as possible. The results are comprehensive, not exhaustive.

The literature on robust estimation is a varied mixture of theoretical and practical proposals. Some comparative studies are available concentrating primarily on asymptotic theory, including the papers of Huber (19 64), Bickel (1965), Gastwirth (1966), Crow and Siddiqui (1967), and Birnbaum and Laska (1967). In addition, various useful small sample studies have been made, based either on Monte Carlo methods or exact moments of order statistics. Among the former we include Leone, Jayachandran, and Eisenstat (1967) and Takeuchi (1969) while the latter include Gastwirth and Cohen (1970) and Filliben (1969).

As do these authors, we address ourselves to the problem of point estimation of location. The significant advantages of this study are size and scope.

The aims of this survey are, by bringing together a wide experience with a variety of concepts, to (i) augment the known properties of robust estimators (ii) develop and study new estimates.

A great variety of some 6 8 estimates were studied, some well known, others developed during this study; some hand-computable, others requiring a great deal of computer time; some suitable for sample size 5, others best suited to sample sizes over 100. In particular, trimmed means, adaptive estimates , Huber estimates , skipped procedures and the Hodges-Lehmann estimate are among the estimates considered. The estimates are outlined in Chapter 2. Computer programs used to implement these procedures are given in Appendix 11.

The diversity of the survey extends to the variety of properties of both large and small samples which were investigated. Variances of the estimates were calculated for samples of 5, 10, 2 0 and 40. Percentage points were calculated. Breakdown bounds were studied. Hampel's (1968) influence, or Tukey's (1970) sensitivity, curves were calculated to study the dependence of estimators on isolated parts of the data. The ease of computation was investigated. A wide variety of sampling situations were considered, some designed deliberately to test weak points of the procedures considered.

Several new asymptotic results were established to evaluate variances and influence curves. While these were not found for all the estimates in this study, a large number are recorded in Chapter 3. Of particular interest are the formulae for skipped procedures and the unpleasant features of the "shorth".

In the course of this study, W. H. Rogers developed Monte Carlo techniques for sampling from a family of distributions proposed by J. W. Tukey (scale mixtures of zero mean Gaussians) which yielded very great accuracy with a moderate number of samples. The theory used in this study is reported in Chapter 4.

In Chapter 5, numerical results are given for a variety of characteristics. In Chapter 6, these results are analyzed in detail, introducing new methods for the summarization of such a large body of data. J. W. Tukey shouldered this responsibility. Brief additional summaries reflecting the varied experience of some of the contributors are included in Chapter 7.

In the appendices, computer programs for the estimates are given and some material related to the study of robust procedures is included. Appendix 11, as mentioned above, contains the estimate programs used in this study. Appendix 12 contains programs and details for the random number generation of the study. New methods for displaying and assessing variances were developed. The general theory and details relevant to particular distributions are given in Appendix 13. Various numerical integration and Monte Carlo sampling schemes were considered for use in this project. Some considerations and resulting developments are included in Appendix 14. Finally in Appendix 15 are outlined some possible extensions for more efficient Monte Carlo methods.

The results of this study are too diverse to summarize at all completely. We can, however, offer the following guidance for a quick exploration, whether or not this is to be followed by careful study: read Chapter 2 as far as you are strongly interested, then do the same for Chapter 3. Read sections 5E and 5F, then sections 6A, 6B, 6C, continuing as far in six as strongly interested. Begin Chapter 7, continuing as long as interested.

ESTIMATES

2A INTRODUCTION AND INDEX OF ESTIMATES

A great variety of some 6 8 estimates have been included in this study. All of these have the property that they estimate the center of a symmetric distribution (assured by constraint (ii) below). Indeed some were designed to optimize some aspect of this estimation procedure. Other estimates were designed under different conditions or with different objectives in mind. The only constraints on the estimates have been:

(i) all estimates must be represented by computable algorithms,

(ii) all estimates must be location and scale invariant in the sense that if each datum is transformed by x -> ax + b then the estimate T = T(x) is similarly transformed by T -> aT + b.

In this chapter we give brief descriptions of the estimates T([??]) in terms of x(1) ≤ x(2) ≤ ... ≤ x(n), the order statistics of a sample [??] = (xl, ..., xn). The estimates are grouped according to type, e.g., trimmed means, M-estimates etc. with cross references being given at the end of each section.

The descriptions given are not to be taken too literally for the more complicated procedures. The reader who desires certainty rather than a very close approximation should refer to the program as given in Appendix 11.

In those cases in which the originator of an estimate is mentioned without a reference it is to be assumed that the procedures were proposed in the course of this study.

To each estimate there corresponds a number and a mnemonic. For example, the mean, estimate 1, is denoted by M, the median, estimate 6, is denoted by 5 0%. The number corresponds roughly to the order...