🚀 go-pugleaf

rossini@biosun4.harvard.edu (Anthony Rossini) writes:

>Correct me if I am wrong, but when you say non-parametric, are you excluding
>semi-parametric models as well?  If you are looking at strictly
>non-parametric estimators, such as estimators for distribution functions,
>then the asymptotics (which sometimes result in funky cube-root things)
>might result in Rao-Blackwell type theorems, but not easily understandable
>ones (at least to me, so far).

I'm not quite sure how to answer this since I have yet to hear a
definition of "semiparametric" that I am wholly happy with.  What I
would like is results for estimation of a finite vector of functionals
of the distribution in a case where the distribution is known to lie
in some infinite-dimensional set.  So the estimand is finite
dimensional but the probability model is not.  If this is what you
mean by semiparametric then that is what I want.  The simplest example
that I can think of would be estimation of the mean of a distribution
for which it is only known that first and second moments exist.  Some
would call this a nonparametric problem, however.

I would also like to be able to handle problems where the probability
model is restricted in a more substantial way by qualitative
restrictions, but not so much as to admit a finite dimensional
parameterization.

An example of the latter form that I have given some thought to is:

Suppose that we observe and i.i.d. sample of (x,y) that are known to
be related by the equaiton y = x'b, where b is a random vector of the
same dimension as x that is assumed to be distributed independently of
x.  Assume only

           y = x'b
           b is independent of x
           E(x'x) and E(b'b) are finite.

These assumptions substantially restrict the distribution of (y,x),
but the model is nevertheless nonparametric.  Now suppose that I want
to find the UMVU estimator for E(b).  This is a problem that I am
actually working on at the moment.

The semiparametric efficiency bounds literature can be used to show
that a generalized least squares estimator (or a feasible generalized
least squares estimator) is asymptotically efficient.  The
Gauss-Markov theorem tells us that the GLS estimator is best among all
_linear_ uniformly unbiased estimators in a finite sample.  But I
don't know of any results showing that the GLS estimator is best among
all uniformly unbiased estimators in a finite sample.

-- Scott
--
T. Scott Thompson              email:  thompson@atlas.socsci.umn.edu
Department of Economics        phone:  (612) 625-0119
University of Minnesota        fax:    (612) 624-0209