In article <[EMAIL PROTECTED]>,
Robert J. MacG. Dawson <[EMAIL PROTECTED]> wrote:
>Herman Rubin wrote:
>> In article <[EMAIL PROTECTED]>,
>> Dave <[EMAIL PROTECTED]> wrote:
>> >Hi, can anyone give me a definition of what parametric statistical
>> >testing is when compared to non-parametric?
>> If a model is given with a finite number of parameters
>> for the underlying distributions and structure, or at
>> worst a finite number of parameters to be estimated,
>> it is called "parametric". Else, it is misnamed
>> "non-parametric"; it should be "infinite parametric"
>> as a proper description of what is to be inferred
>> involves an infinite number of parameters.
> Would you like to give us an example of such a "proper description"?
><grin>
Consider the estimation of a density or a spectral
density. Most of the approaches use a method to produce a
function. Now one might think that specifying a function
does not specify any parameters, but it actually specifies
infinitely many. In fact, insisting that data are normal
specifies infinitely many parameters.
>Also, surely it would be incorrect to infer a model with more
>parameters than one had data.
Wrong, as can be seen from the above. It is often a major
error to believe that a low-parametric model is "correct";
when one approaches decision making from the rational, or
consistent, point of view, the logical choice is that all
models are simultaneously present, and the question is about
which will give better results.
> I know what Herman is getting at, but I don't agree with the details.
>The point of a non-parametric model is that one does *not* attempt to
>infer parameter values.
One might not attempt to, but one cannot avoid it.
> One might also ask, what exactly is a parameter?
A parameter is anything which can be computed from full
knowledge of the exact model. It is identified if one
can compute it from the probability distribution of the
observed data.
It's usual to suppose
>that (for instance) the mean is always a parameter; but is it, if one
>doesn't have the rest of a parametric model?
Of course it is.
Conversely, if one has a
>parametric model, the median is usually a valid parameter. (If the
>models in the family are symmetric, it is the same as the mean, and the
>sample mean and sample median are both valid estimators.)
Parameters have nothing to do with the estimation process.
It is one of the jobs of the statistician to help the user
estimate parameters from the user's model and the data.
> One might, I think, argue that (for instance) the t test, done on a
>sample large enough that one is not concerned about close approximation
>to normality, is in fact nonparametric, in that no specific parametric
>model is ever assumed.
This is a different use of non-parametric. Even here, there
are parametric assumptions; the mean and variance must exist,
and the fourth moments not be too large (this can depend on
the sample size). Such tests as the Kolmogorov-Smirnov are
non-parametric in the class of all continuous distributions;
such tests are the only procedures which can really be called
by the non-parametric title.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
[EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558
.
.
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