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UNIVERSITY OF WISCONSIN - MADISON MATHEMATICS RESEARCH CENTER ESTIMATION OF A MIXING DISTRIBUTION FUNCTION

IP.com Disclosure Number: IPCOM000128710D
Original Publication Date: 1976-Dec-31
Included in the Prior Art Database: 2005-Sep-16
Document File: 20 page(s) / 43K

Publishing Venue

Software Patent Institute

Related People

R. Blum: AUTHOR [+3]

Abstract

BSTRACT

This text was extracted from a PDF file.
This is the abbreviated version, containing approximately 14% of the total text.

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

UNIVERSITY OF WISCONSIN - MADISON MATHEMATICS RESEARCH CENTER ESTIMATION OF A MIXING DISTRIBUTION FUNCTION

R. Blum and V. Susarla

Technical Summary Report # 1690 October 1976

BSTRACT

Let

(Equation Omitted)

an interval, be a family of univariate probability densities (wrt Lebesgue measure) on an interval I . First. a necessary and sufficient condition is proved for a to be identifiable

whenever

(Equation Omitted)

the class of continuous functions on J vanishing

(Equation Omitted)

G is a G-mixture of the densities in 67 with G unknown, an estimator G n based on f G and

(Equation Omitted)

is provided such that G n --!A~' G under certain conditions on

(Equation Omitted)

random variables from f . an estimator G is provided such that .

(Equation Omitted)

almost surely under bertain conditions on a and G . Furthermore, it is shown that If

(Equation Omitted)

with rates like

(Equation Omitted)

under certain conditions on the density estimator f G (x) involved in the definition of G n . The conditions of various theorems are verified in the case of location parameter and scale parameter families of densities.

University of Wisconsin Page 1 Dec 31, 1976

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UNIVERSITY OF WISCONSIN - MADISON MATHEMATICS RESEARCH CENTER ESTIMATION OF A MIXING DISTRIBUTION

FUNCTION

AMS(MOS) Subject Classification - 60F051 62099.

Key Words - Identifiability, weak convergence, mixing distribution, empirical Bayes. Work Unit Number4 - Probability, Statistics, and Combinatorics

Sponsored by: 1) The United States Army under Contract Number DAAG29-75-C-0024; 2) Grant from the Graduate School of the University of Wisconsin- Milwaukee; 3) HEW, NIET Grant I-RO-l-GM 23129-01; 4) N,ational Science Foundation Grant MCS76-05952. .

Introduction and summary.

Let f be a Borel measurable function from

(Equation Omitted)

such that

(Equation Omitted)

for each 0 in J where I and I j are intervals contained in

(Equation Omitted)

and I be the collections of sections of f with the first coordinate (in 1) and the second coordinate (in J) fixed respectively. For a probability distribution function G on let

(Equation Omitted)

We provide an equivalent condition for the identi,fi ability of a (for the definition of identifiability, see (Al)) in Section 2. In Section 3, we consider the problem of estimating G in terms of f G and To obtain an estimate G n of G, we solve a system of equalities and inequalities and then show that G n converges weakly to ( w --) G under some con- ditions on 6 . If G , and f G are unknown, but iid random variables

(Equation Omitted)

... are observable (this is the standard empirical Bayes situation of Robbins [4] described in Section 4), then we construct (in Section 4)

Sponsored by-.

1) The United States Army under Contract: Number DA.AG29-75-C-0024; 2) Grant from the Graduate School of the University of Wisconsin- Milwaukee; 3) HEW, NIH Grant 1-RO-1-GM 23129-01;

4) National Science Foun...