🚀 go-pugleaf

In article <WVENABLE.92Sep12234508@algona.stats.adelaide.edu.au> wvenable@algona.stats.adelaide.edu.au (Bill Venables) writes:
>>>>>> "Charles" == Charles Packer <packer@amarna.gsfc.nasa.gov> writes:

>Charles> Why not use a traditional chi-square test?

>I can think of two possible reasons:

>1. It requires an arbitrary partition of the range into panels, *before*
>   the sample comes to hand.  (In fact it really does not test normality as
>   such, but rather that the grouped sample distribution agrees with a
>   similarly grouped normal.)  Arbitrariness always comes at some cost.

>2. In seeking to get some power against a very wide class of alternatives
>   it manages only to achieve low power against any subclass, including the
>   subclass of practically important alternatives.  In this sense it is not
>   well focused enough.

>[BTW I would be interested in a Bayesian reaction to this question.  It always
>seemed to me that tests of fit could be rather an embarrassment to a Bayesian.]

Even from the classical standpoint, the traditional chi-squared test is 
wrong.  The distribution of the chi-squared statistic should not have the
full reduction in degrees of freedom for estimated parameters.  This was
proved by Chernoff and Lehmann in 1953 for fixed partitions, and by A. R. Roy
in his dissertation under me in 1954 for data-defined partitions; the theory
using "good" estimators under the null hypothesis shows that it does not 
matter how the partitions are defined.  Some of the papers of D. H. Moore
about 1970 provide a readable summary.

The power is VERY low against any reasonable alternative.  The reason for
this is that the test fails to consider that adjacent intervals are likely
to differ from the null in the same manner.  Such tests as the Kolmogoroff-
Smirnoff or Kuiper do a far better job.

For a discussion of testing from a parametric robust Bayesian viewpoint, 
readers may wish to look at my paper with Sethuraman in Sankhya, 1965, on
the subject.  For at least one "real" problem, that of looking for concentrated
contamination to a distribution, this can be approximately carried out in the
infinite-dimensional version.  But one thing to keep in mind--the higher the
dimension, the more important the assumptions.
-- 
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)