# MINIMAZ ESTIMATION OF THE MEAN OF A STOCHASTIC PROCESS

Original Publication Date: 1978-Dec-31

Included in the Prior Art Database: 2005-Sep-15

## Publishing Venue

Software Patent Institute

## Related People

Lawrence Peele: AUTHOR [+3]

## Abstract

Let I be a set of real numbers and let JY(t): t e I) be a real stochastic process whose covariance kernel is known and is posi- q tive definite. The mean of [Equation ommitted] are known functions on I and the coefficients [Equation ommitted] 0 q are unknown. A minimax mean-estimation procedure involv- q 2 ing the assumption that iLl e. is small is shown to be a balance 1 between the usual least squares mean-estimation procedure and a smallness constraint on the estimates of [Equation ommitted]

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__Page 1 of 9__THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

**MINIMAZ ESTIMATION OF THE MEAN OF A STOCHASTIC PROCESS **

By Lawrence Peele, old Dominion University

**ABSTRACT **

Let I be a set of real numbers and let JY(t): t e I) be a real

stochastic process whose covariance kernel is known and is posi- q tive definite. The mean of

(Equation Omitted)

are known functions on I and the coefficients

(Equation Omitted)

0 q are unknown. A minimax mean-estimation procedure involv- q 2 ing the assumption that iLl

e. is small is shown to be a balance 1 between the usual least squares mean-estimation
procedure and a smallness constraint on the estimates of

(Equation Omitted)

**1. INTRODUCTION A14D NOTATION **

Let I be a set of real numbers and let

(Equation Omitted)

be a real

stochastic process of the form

(Equation Omitted)

where

(Equation Omitted)

are linearly independent, known functions on I and elf 6 21 0 q are unknown constants. The process

(Equation Omitted)

is assumed to have mean zero and known, positive definite covariance kernel k.

Old Dominion University Page 1 Dec 31, 1978

__Page 2 of 9__MINIMAZ ESTIMATION OF THE MEAN OF A STOCHASTIC PROCESS

Let tit t 21 t n e I with q :5 n, and let

(Equation Omitted)

Suppose that (1.2) rank (Z) = q so that for

(Equation Omitted)

there exists a unique minimum vari-

ance unbiased linear (MVUL) estimator n A (1.3)

(Equation Omitted)

See Parzen (1961) for a discussion of regression analysis for a stochastic process. Let M denote the collection of all lin- ear combinations of

(Equation Omitted)

and for

(Equation Omitted)

(Condition (1.2) implies that each m c M corresponds to a unique m e M.) Given the obser- vations

(Equation Omitted)

the MVUL estimate for 0 is

(Equation Omitted)

if

(Equation Omitted)

then it is well known that among

(Equation Omitted)

minimizes

(Equation Omitted)

where

(Equation Omitted)

is the n x n covariance matrix for

(Equation Omitted)

Old Dominion University Page 2 Dec 31, 1978

__Page 3 of 9__MINIMAZ ESTIMATION OF THE MEAN OF A STOCHASTIC PROCESS

That is, this MVUL esti- mate m* for the mean of

(Equation Omitted)

is the covariance-weighted least squares fit to the observations (1.4).

Suppose that instead of just using least squares fitting, one wishes to force the estimates for the coefficients elf e 2'

0 q to be small. if 0 is some positive constant, one might wish to fit the observations (1.4) by choosing m" c M to minimize

(Equation Omitted)

among

(Equation Omitted)

This choice of m" would be a balance between covariance-weighted least squares fitting and a desire for small coefficient estimates. It will be shown that the solu-tion to a particular minimax mean-estimation problem is to choose m c M to minimize (1.7).

**2. BACKGROUND MATERIAL **

This paper is related to a paper by Peele and Kimeldorf (1977) that extendssome prediction results by Kimeldorf and Wahba (1970) and examines certain MVUL and minimax mean-estima- tion problems associated with a stochastic process. Some predic-tion and...