# Robust Estimates of Location: Survey and Advances

D. F. ANDREWS
P. J. BICKEL
F. R. HAMPEL
P. J. HUBER
W. H. ROGERS
J. W. TUKEY
Pages: 386
https://www.jstor.org/stable/j.ctt13x12sw

1. Front Matter
(pp. i-iv)
2. PREFACE
(pp. v-vi)
(pp. vii-x)
4. 1. INTRODUCTION
(pp. 1-2)

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...

5. 2. ESTIMATES
(pp. 3-28)

A great variety of some 68 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...

6. 3. ASYMPTOTIC CHARACTERISTICS OF THE ESTIMATES
(pp. 29-54)

In this section various asymptotic characteristics of some of the procedures introduced in Chapter 2 will be stated. Where the formulae are novel, a brief semirigorous derivation will be given. In all other cases references to the literature are provided.

The following notation will be used throughout. Thesamplex₁,...,xnis a set of independent and identically distributed observations from a parent distribution F which is assumed continuous. If F is absolutely continuous, its density is denoted by f. Theempirical distributionof the sample will be denoted by Fnand as usual defined by

${F_N}(t) = \frac{1}{n}\sum\limits_{i = 1}^n I [{x_i} \le t]$

The order statistics, denoted by...

7. 4. FINITE-SAMPLE CALCULATIONS
(pp. 55-63)

The main body of finite-sample results is derived from four basic quantities or properties of the estimators:

1. Average (or bias)

2. Variance (or mean square)

3. Covariance between pairs of estimators

4. Percent-points (or quantiles).

These quantities have been calculated for 65 estimators and 30 different sampling “situations” (parent distributions and sample size) covering the Normal distribution and longer-tailed alternatives. Sample sizes ranged from 5 to 40.

One simple but inefficient approach to such a problem is to regard the sampling situation and the estimator as black boxes. The numbers which the boxes produce can be processed through the...

8. 5. PROPERTIES
(pp. 64-115)

This chapter summarizes the numerical results obtained in the study. The properties of all the estimates studied are recorded; there is no attempt in this chapter to draw conclusions or dwell on the implications of particular results.

The numbers are frequently recorded to precision greater than is warranted for individual results (average 1.8 decimal digits). However since all estimates were evaluated on the same samples, within each situation, relative comparisons may be meaningful when these occur in the third decimal digit.

A brief description of each table is given in Exhibit 5-1....

9. 6. A DETAILED ANALYSIS OF THE VARIANCES
(pp. 116-221)

This chapter endeavors to analyze a substantial part of the detailed results, which include, in addition to what has been presented earlier:

- covariances between every pair of estimates for every situation.

- indications of variance for unbiased linear combinations of every pair of estimates for every situation. (Coefficients step by 1/4, providing three internal combinations. One external combination is available at each end.)

There has been no attempt to analyze covariances and linear combinations in any detail. Selected instances have been looked at with interesting results. We have uncovered enough leads for progress by altering estimates to make an attempt at...

10. 7. GENERAL DISCUSSION
(pp. 222-260)
J. W. Tukey, D. F. Andrews, F. Hampel, P. Huber and P. J. Bickel

Those who have participated actively in the study here reported are convinced that both science and art have been advanced significantly. It is time to try to make this clear to all -- and, particularly, to make clear the diversity of ways of advance.

We have learned that hampels are extremely promising, and deserve both extension and tuning.

We have learned that sitsteps do very well in view of the limited computational effort required, and deserve careful tuning at both ends. The use of the sit mean as the start of the one-step estimate demands inquiry.

We have learned that...

11. 11. PROGRAMS OF THE ESTIMATES
(pp. 261-305)
12. 12. RANDOM NUMBER GENERATION - DETAILS
(pp. 306-309)
13. 13. DUAL-CRITERION PROBLEMS IN ESTIMATION
(pp. 310-333)
14. 14. INTEGRATION FORMULAS, MORE OR LESS REPLACEMENT FOR MONTE CARLO
(pp. 334-348)
15. 15. MONTE CARLO FOR CONTAMINATED GAUSSIANS
(pp. 349-368)
16. REFERENCES
(pp. 369-373)