From: maj@waikato.ac.nz
Newsgroups: sci.math.stat
Subject: Re: robust location estimators
Date: 11 Oct 92 12:18:34 +1300
Organization: University of Waikato, Hamilton, New Zealand

In article <1992Oct2.084819.11178@waikato.ac.nz>, maj@waikato.ac.nz writes:
> In article <BvGFAu.GEv@mentor.cc.purdue.edu>, hrubin@pop.stat.purdue.edu (Herman Rubin) writes:
> 
>> 
> [stuff and quotes deleted]
> 
>> Robustness cannot be defined in a formal manner to be a precise concept.
>> The definition I like is
>> 
>> 	The robustness of a procedure is the extent to which its 
>> 	properties do not depend on those assumptions which one
>> 	does not wish to make.
>> -- 
>> Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
>> Phone: (317)494-6054
>> hrubin@pop.stat.purdue.edu (Internet, bitnet)  
>> {purdue,pur-ee}!pop.stat!hrubin(UUCP)
> 
> This is a good definition of what one wants, but I dispute that
> robustness cannot be given a formal definition. I'll try:
> 
>    A statistical functional is robust iff it is continuous.
> 
> Continuous in what topology?   you may ask. *That* I wont answer,
> that's were the vagueness of the concept comes in.


An email correspondence with Peter Hamer leads me to think that I was perhaps a
little on the terse side in my earlier posting, he writes:
      
      >       Take the distance between two cdfs F and G to
      >       be given by, say, the sup norm.
      >
      >       It is easy to see that within any epsilon of
      >       F there are distributions with arbitrarily
      >       different mean.

      I don't know how you intended this to be interpreted. I seems
      very much like saying that applied statistics is impossible;
      as no real data is *known* to come from a precisely specified
      distribution.

      All applied statistical procedures must be applicable to distributions
      `close to' the one nominally assumed, and you seem to be saying that
      this is impossible.
 

      Regards, Peter (pgh@bnr.co.uk)
 
 

[My reply may be of interest to other readers of this group.]

Rather than try to persuade you about the correctness of my
remarks I have looked up a few references. Have a look at

Staudte & Sheather 'Robust Estimation and Testing' Wiley 1990
Section 3.2.4 pp65-67

Hampel, F.R. 'A General Qualitative definition of Robustness'
Ann Math Stat v42, 1887-1896, (1971)
[esp. Theorem 1, p1891]

Huber, P.J. 'Robust Statistical Procedures'(1977)  #27 in
CBMS-NSF series.
[esp. Chapter 2 and first part of Ch 3. I prefer this to
the more elaborate treatment in his 1981 book.]

Huber and Hampel work in the full generality with statistics
understood as sequences of functionals. However the main
points are unchanged and more easily understood by considering
only functionals.

Example: the divisor n-1 standard deviation can be represented
         as a sequence of functionals, one for each sample
                  size.
                  The divisor n standard deviation can be represented
                  as a single functional with no need to involve
                  sample size.
                  
The discontinuity of the sample mean as a function from a
space of cdfs to the reals poses no real problem to
*applied* statistics because in practice we do not really use
the mean by itself: we actually employ data inspection 
followed by transformations and/or outlier deletion. In reality
the robust perspective is more of a threat to traditional
*mathematical* statistics with its highly developed theory
about the behaviour of relatively simple statistics at
precisely specified models, something that does not really
model modern applied statistical practice in the least.            

       

-- 
Murray A. Jorgensen [ maj@waikato.ac.nz ]    University of Waikato
Department of Mathematics and Statistics     Hamilton, New Zealand      
__________________________________________________________________
'Tis the song of the Jubjub! the proof is complete,
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