Herman Rubin wrote:
> >> 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.
and later
> 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.
and
> A parameter is anything which can be computed from full
> knowledge of the exact model.
The word "parameter" appears to be being used here in
two mutually incompatible ways. The first, earlier quote is
consistent with what I would have taken as the usual definition
of "parameter", namely, a variable indexing a family of
functions/distributions/what-have-you. The concept (in this
sense) has no meaning outside this context; asking in the
abstract "is the mean a parameter?" is like asking "is the
group D4 isomorphic?" or "is (0,1) a local maximum"?
(You know the joke: examiner, "Which of these three groups
are isomorphic?" student "The first two aren't but I think the third
one is.")
Thus, for instance, the mean can be a parameter of the
N(mu, sigma^2) family, the N(mu, 1^2) family, and the U[0,A]
family of distributions. It cannot be a parameter of the
N(0,sigma^2) family or the U[-A,A] family - despite the fact
that it can be calculated from the model. It and the third
quartile together are parameters of the N(mu,sigma^2) family,
the U[A,B] family, but not of any other family of distributions
given above.
-Robert Dawson
.
.
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