From: maj@waikato.ac.nz
Newsgroups: sci.math.stat
Subject: Robustness
Date: 16 Oct 92 08:41:17 +1300
Organization: University of Waikato, Hamilton, New Zealand

           I wrote :
           ~~~~~~~~~
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.]

[stuff deleted]
               
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.     

          Herman Rubin responded [and my riposte is interpolated]:
          ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This attempt to avoid the problem does not succeed.  Any crass attempt
to delete outliers messes up things but good; in most situations, I am
much more inclined to go along with the long tails than with the total
damage to the analysis which these produce, especially if there are
explanatory variables.

           I am not quite sure what is meant by 'going along with' long
           tails. Naturally outlier rejection methodology which potentially
           draws on information about the problem other than that in the
           sample will be difficult to model asymptotically, but it does
           not lose legitimacy because of this. Robust statistics which
           downweight outliers _can_ be studied asymptotically, and
           although rarely used in practice can be thought of as a 
           formalization of existing good applied statistical practice
           based on examination of outlying and influential points.

  As for transformations, unless one is looking
only at the distribution of a single random variable, they should NEVER
be used, as they are almost certain to destroy the model. 

          So they destroy the model! What do I care? What did the
          model ever do for me? Seriously the model is only a construct
          used to help us understand the data. If we were ever fortunate
          enough to have sample size tending to infinity and disk quota
          doing the same we would inevitably observe fine structure that
          would make us want to refine and elaborate the model. It is
          not the _model_ that is sacrosanct, it is the _data_.

          By the way, I seem to remember that Box is not averse to the
          odd transformation.

 The observation
that the model is not exactly correct is quite appropriate, but while
it is possible to show that robustness in the sense I have given, which
is a slight extension of the original definition of Box, is quite possible,
robustness in the Huber-Hampel sense is so rare for reasonable models as
to be a fatuity.

          I could counter that the use of nonrobust statistics like
          the mean and standard deviation without the removal of
          "obvious errors" [not quoting h.r.] is so rare as to be
          a fatuity.

The laws of large numbers are useful robustness theorems.  The Central
Limit Theorem is a robustness theorem.  For regression, the Gauss-Markov
Theorem is a robustness theorem of the important type of practical
situations not covered by the use of continuity.

          Sure they are robustness theorems. The only problem with
          them lies in the strength of their hypotheses, not with
          the strength of their conclusions.

  Any time that it
is shown that normality is not needed, but that a few moments suffice,
one has a robustness result which is not covered by the Huber-Hampel
definition.

          I am not sure that this kind of situation is not covered
          by the H-H definition. Why not define a metric by

          $$d_k(F,G)=\mathop{\sup }\limits_{\nu \in \Re }
          \int_{-\infty }^\infty  {|x-\nu |^kd|F-G|}$$
          [OK, OK, OK, the guts of it is the integral of
          abs(x - something)**k w.r.t abs(F-G).]

          It seems to me that continuity in this sort of metric for
          k = 2, 3, or 4, say, expresses the kind of robustness that
          Dr Rubin is referring to.
          
The well-known asymptotic properties of maximum likelihood and Bayes
estimates are among this class.  The mean may be the parameter of interest;
there is no Huber-Hampel robust estimator of it.

BTW, most simulations assume symmetry.  This is a far stronger assumption
than merely having a few moments.

          Symmetry is only a convenience so that comparative simulations
          can be seen to be comparing like with like. Every functional
          statistic is Fisher-consistent for its value at the
          population distribution. Usually it is only a matter of
          convenience what statistic one adopts. For example the
          sign test of the hypothesis that the median of the differences
          is zero is often used as a nonparametric substitute for the
          paired t test.
--
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)
-- 
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,
if only I've stated it thrice.'
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