Eric B. Hall

Some aspects of fusion in estimation theory

The problem of fusing or combining various estimates to obtain a single good estimate is investigated. The problem of fusion in estimation theory is addressed, and several examples using common distributions in which virtually any method of fusion would be useless in approximating the random variable of interest are presented. A theorem which, for a very general situation, shows that fusion resulting in an almost surely exact approximation is always possible is presented. In particular, this result addresses the situation in which the data consists of the random variable of interest corrupted by additive Gaussian noise and the random variable of interest could be any second-order random variable. Finally, an example which illustrates the utility of this result is presented. >

On optimal estimation with respect to a large family of cost functions

The authors consider the problem of optimal estimation of a random variable X based on an observation denoted by a random vector Y. A commonly encountered problem involves estimating X via h(Y) so as to minimize E( Phi (X-h(Y))), where h is Borel measurable and Phi is a Borel measurable cost function chosen to adequately reflect the fidelity demands of the problem under consideration. The authors place a mild condition on the regular conditional distribution of X given sigma (Y) that ensures that E( Phi (X-h(Y))) is minimized for any cost function Phi that is nonnegative, even and convex. In addition, it is shown that given any Borel measurable function g: R to R, there exist random variables X and Y possessing a joint density function such that E(X mod Y=y)=g(y) almost everywhere with respect to Lebesgue measure. >

A curiosum concerning discrete time convolution

It is shown that the discrete-time convolution of two absolutely summable nowhere-zero sequences may be identically equal to zero. >

A RESULT ON MULTIDIMENSIONAL QUANTIZATION

For any integer N > 1 , a probability space, a Gaussian random vector X defined on the space with a positive definite covariance matrix, and an N-level quantizer Q are presented such that the random vector Q(X) takes on each of the N values in its range with equal probability and such that X and Q(X) are independent.

A note on the distribution of the determinant of a random matrix

An analysis of a probability density function of the determinant of a random matrix is presented, and an oversight in an earlier paper on this subject is noted.

Importance sampling via a simulacrum

Abstract A Monte Carlo variance reduction technique known as “importance sampling” has recently been applied to many problems in data communications. This technique holds the promise of offering vast improvements to traditional Monte Carlo methods. An overview of importance sampling applied to the calculation of tail probabilities is presented, as well as examples for which some popular approaches to importance sampling fail to work. New techniques for the calculation of the resulting variances are introduced, as well as a new approach to importance sampling which offers the promise of substantial variance reduction over previous techniques.

A comment on the finite memory of stochastic processes

It is shown that a proposed concept of finite memory for a zero-mean strictly stationary stochastic process results in a stochastic process of random variables each of which is almost surely equal to zero. >

Some comments on filtering in estimation theory

Mathematical oversights that often seem to accompany efforts in filtering which arise in the context of estimation theory are considered. In particular, problems associated with such popular topics as convolution, Kalman filtering, and optimal nonlinear filtering are considered. A version of the extended Kalman filter is discussed.<<ETX>>

On mutual independence

For any integer N > 1, we construct a probability space and N standard Gaussian random variables X1, X2,..., XN such that P(Xi [set membership, variant] Ai for each i [set membership, variant] I) = [Pi]i[set membership, variant]IP(Xi[set membership, variant]Ai) holds whenever the sets in {Ai: i [set membership, variant] I} are Borel sets but does not hold for all sets {Ai: i [set membership, variant] I} for which the indicated probabilities are well defined where I is any subset of {1, 2,..., N} having cardinality at least two. We then extend this result to the case where the random variables can have any diffuse distributions.

Estimation of a random variable based on multidimensional data

Several aspects of the mean square estimation of a second-order random variable based upon elements from a random field are considered. The role of the underlying probability space is stressed. Numerous examples are presented that point out many of the subtleties associated with this endeavor.<<ETX>>