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Licensed Unlicensed Requires Authentication Published by De Gruyter February 24, 2011

Use of the global implicit function theorem to induce singular conditional distributions on surfaces in n dimensions: Part II

Donald Bamber EMAIL logo , I. R. Goodman , Arjun K. Gupta and Hung T. Nguyen
From the journal

Abstract

This paper stems from previous work of certain of the authors, where the issue of inducing distributions on lower dimensional spaces arose as a natural outgrowth of the main goal: the estimation of conditional probabilities, given other partially specified conditional probabilities as a premise set in a probability logic framework. This paper is concerned with the following problem. Let 1 ≤ m < n be fixed positive integers, some open domain, and h : a function yielding a full partitioning of D into a family, denoted M(h), of lower-dimensional surfaces/manifolds via inverse mapping h–1 as D = ∪ M(h), where M(h) = d {h–1(t) : t in range(h)}, noting each h–1(t) can also be considered the solution set of all X in D of the simultaneous equations h(X) = t . Let X be a random vector (rv) over D having a probability density function (pdf) ƒ. Then, if we add sufficient smoothness conditions concerning the behavior of h (continuous differentiability, full rank Jacobian matrix dh(X)/dX over D, etc.), can an explicit elementary approach be found for inducing from the full absolutely continuous distribution of X over D a necessarily singular distribution for X restricted to be over M(h) that satisfies a list of natural desirable properties? More generally, for fixed positive integer r, we can pose a similar question concerning rv ψ (X), when ψ : is some bounded a.e. continuous function, not necessarily admitting a pdf.

Received: 2009-06-01
Accepted: 2009-12-12
Published Online: 2011-02-24
Published in Print: 2011-March

© de Gruyter 2011

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