Abstract
The purpose of this paper is to propose a method of constructing exponential
families of Hilbert manifold, on which estimation theory can be built. Although
there have been works on infinite dimensional exponential families of Banach manifolds
(Pistone and Sempi, 1995; Gibilisco and Pistone, 1998; Pistone and Rogantin,
1999), they are not appropriate to discuss statistical estimation with finite number
of samples; the likelihood function with finite samples is not continuous on the
manifold.
In this paper we use a reproducing kernel Hilbert space as a functional space for
constructing an exponential manifold. A reproducing kernel Hilbert space is dened as a Hilbert space of functions such that evaluation of a function at an arbitrary
point is a continuous functional on the Hilbert space. Since we can discuss the
value of a function with this space, it is very natural to use a manifold associated
with a reproducing kernel Hilbert space as a basis of estimation theory.
We focus on the maximum likelihood estimation (MLE) with the exponential
manifold of a reproducing kernel Hilbert space. As in many non-parametric estimation
methods, straightforward extension of MLE to an infinite dimensional
exponential manifold suffers the problem of ill-posedness caused by the fact that
the estimator should be chosen from the infinite dimensional space with only finite
number of constraints given by the data. To solve this problem, a pseudo-maximum
likelihood method is proposed by restricting the infinite dimensional manifold to
a series of finite dimensional submanifolds, which enlarge as the number of samples
increases. Some asymptotic results in the limit of infinite samples are shown,
including the consistency of the pseudo-MLE.
