Stochastic approximation and selection of a conditional extermal function in continuum

Abstract

Stochastic approximation aims to estimate efficiently level crossing or extermal points of a nonparametric regression function in a parametric or nonparametric manner. For bundle strength of parallel filaments (with stochastic length, cross section etc.,) and in some other problems, one has an extended statistical functional where ψ is a suitable function and G(x|y) stands for a conditional distribution function of X, the dependent variate, given the independent variate Y=y. For a compact set C chosen from extraneous considerations, the problem of interest is to locate an optimal y⁰∊C, in the sense that . Nonparametric estimation of the conditional distribution G(.|y), nonlinear nature of the functional θ(·), and the fact that y is in continuum in C add complexity to possible statistical resolutions and create impasses for a direct application of the well known Bechhofer-Gupta-Sobel methodology. General asympotics for the selection procedures arising in this context are considered with due emphasis on a stochatic iteration scheme.

Keywords:

• Bundle strength of filaments
• Concomitants of order statistics
• Consistency
• Nearest nighbor method
• Sequential rule
• Stochastic iteration

Additional information

Notes on contributors

Pranab K. Sen

Zhou Zhenwei