Publication Date

5-1-1974

Abstract

In a recent paper Stoughton Bell and William J. Zimmer defined three distinct types of conditional independence; which they call pointwise, intervalwise and strong; of two random variables given a third. They considered these and four closely related independence properties, and they examined the relationships among the 128 (=23+4) Boolean combinations. Bell and Zimmer also considered the kinds of assumptions in applied research that would give rise to each type of conditional independence.

I have searched published texts and papers in probabil­ity and statistics for the three types of conditional inde­pendence. The type considered by Alfred Renyi [Foundations of Probability (San Francisco: Holden-Day, 1970), pp. 103 - 104] differed from all three types given by Bell and Zimmer. No other author considers a type other than pointwise.

A major result of this dissertation is that strong conditional independence can be embedded in the structure of Renyi, and the pointwise type is equivalent to that given by M. Loeve [Probability Theory (New York: Van Nostrand, 1963), pp. 3 51, 364]. As a consequence of that result, much of the literature of conditioning that is expressed in terms of abstract probability spaces can be studied, in the manner of Bell and Zimmer, on Borel spaces. None of the essential structure is lost in working with cumulative distribution functions rather than abstract probability measures.

In this dissertation I define, for constant regression (which I call essential independence) and zero correlation, analogues of Bell and Zimmer's seven conditions. I then extend and modify their results.

A salient point of this dissertation is that there are several seemingly distinct but logically equivalent ways of expressing constant regression. This equivalence connects local and global (averaging) properties. There is no similar observation to be made for zero correlation. On the other hand, there is an analogous, and even richer, collection of equivalent conditions for independence. This fact has been exploited by Bell and Zimmer and is adapted here.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

William John Zimmer

Second Committee Member

Stoughton Bell

Third Committee Member

Julius Rubin Blum

Language

English

Document Type

Dissertation

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