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Started by T. Scott Thompson
Tue, 29 Sep 1992 19:15
UMVU estimation for infinite-dimension models
Author: T. Scott Thompson
Date: Tue, 29 Sep 1992 19:15
Date: Tue, 29 Sep 1992 19:15
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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
Re: UMVU estimation for infinite-dimension models
Author: Anthony Rossini
Date: Tue, 29 Sep 1992 21:07
Date: Tue, 29 Sep 1992 21:07
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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). -tony -- Anthony Rossini - rossini@biostat.harvard.edu Department of Biostatistics, Harvard School of Public Health 677 Huntington Ave, Boston MA 02115 617-432-1056
Re: UMVU estimation for infinite-dimension models
Author: T. Scott Thompson
Date: Wed, 30 Sep 1992 16:23
Date: Wed, 30 Sep 1992 16:23
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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
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