🚀 go-pugleaf

Perhaps one of the theorists out there can address this question:

The literature on estimation given i.i.d. sampling and a parametric
probability models includes three important results:

(1) The Cramer-Rao lower bound on the asymptotic variance of any
"regular" estimator.

(2) The Cramer-Rao lower bound on the variance of a uniformly unbiased
estimator.

(3) If T is a complete sufficient statistic (with respect to a given
probability model) and b is a uniformly unbiased estimator of a given
parameter of interest then there is an essentially unique minimum
variance unbiased estimator given by E(b|T).

I am interested in whether/how these generalize to nonparametric
probability models.  For example, the literature on semiparametric
efficiency bounds clearly provides a generalization of (1) to
semiparametric and nonparametric probability models.  I am not aware
of any nonparametric forms of the finite sample results (2) or (3),
however.

Do any of you know of any pointers/results on this?

The asymptotic generalization of (1) to nonparametric models is
roughly equivalent to looking for a worst case parametric submodel.
It occurs to me that a similar approach might yield generalizations of
(2) and (3).

To give a concrete example of the kind of questions that I have in
mind, suppose that I have an i.i.d. sample and an estimator of the
U-statistic form that is an unbiased estimator of some parameter of
interest.  Suppose that the true distribution from which the data were
sampled is known to lie in some set of probabilities, possibly
restricted in some qualitative fashion, but not so much as to admit a
finite-dimensional parameterization.  How might I check whether or not
my estimator is UMVU?

I will summarize replies.  Thank you in advance.
--
T. Scott Thompson              email:  thompson@atlas.socsci.umn.edu
Department of Economics        phone:  (612) 625-0119
University of Minnesota        fax:    (612) 624-0209