>>There are some procedures which are non-parametric within the
>>class of all continuous distributions. The other so-called
>>non-parametric procedures are really infinite-parametric.
>>This difference is important; if one realizes that one is
>>looking for an infinite number of parameters, the behavior
>>is likely to be quite different than if one thinks that no
>>parameters are involved.
>Could you expand on this, Herman? Sounds interesting, but I'm
>not sure I understand exactly.
Let me give an example. A standard problem is to estimate the spectral
density function for a stationary time series with integer time points.
The density function f is a symmetric function on (-pi, pi).
But the density function is the sum of c_k exp(ikt), which is its Fourier
series. The c_k, except for a factor of 2pi, are the lagged covariances.
This has been treated as a "non-parametric" problem from the beginning,
although the procedure is often stated in terms of the estimation of
the c_k, which are an infinite set of parameters. It is possible to
get robust pseudo-Bayesian procedures (these are not obtained by using
posteriors, but by minimizing the prior expected value of risk among
a class of procedures) which, under fairly reasonable conditions, will
do asymptotically as well as any of a rather large family of presently
used procedures, even if one does not know which one to use. I cannot
see how make this approach without in effect displaying the infinite
set of parameters.
>BTW, whenever I see "non-parametric" inference for medians, I
>get suspicious: Sometimes standard NP tests like Mann-Whitney
>are presented as testing for zero median difference, when the null
>hypothesis really is identical distributions. Also, "NP
>confidence limits for the median difference" based on the M-W
>test actually assume that the distributions have the same shape,
>an assumption not much less restrictive than those of the
>parametric counterparts, but more dangerous because the "NP"
>prefix dissuades users from checking the assumptions.
Your comments are right on. If you are interested in testing
for the equality of medians, test that, and not something else.
And you are right on the assumptions for the "NP confidence
limits." There are reasonable parametric procedures for the
testing, which do not have precisely computable levels, but
which are at least appropriate tools.
--
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@stat.purdue.edu (Internet, bitnet)
{purdue,pur-ee}!snap.stat!hrubin(UUCP)