Document Type


Date of Award



Stochastic analysis, Approximation theory

Degree Name

Doctor of Philosophy (PhD)


Mathematical Sciences

First Advisor

David A. Edwards

Second Advisor

David L. Hanson

Third Advisor

Eugene M. Klimko


In many statistical experiments one wishes to obtain a desirable “level of response” corresponding to some “level of treatment.” The response to a given treatment, however, is usually random, and the best one hopes for is to locate the level of treatment that produces the desired response on the average. The mathematical formulation of the problem is as follows.

For every level of treatment x, which we assume to be numerical and refer to as an “observation point,” the response (“observation”) y at x is a random variable on some probability space with distribution function Fx and mean m(x) < ∞. Thus m defines a regression function. One wishes to locate a point θ such that m(θ) = α1, where α1 is the desired level of response. A stochastic approximation is a sequential estimation procedure where future observation points are determined on the basis of past information. The two most-discussed procedures for the problem described are the Robbins-Monro (R-M) procedure and the up-and-down method of Dixon and Mood.