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

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 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 a fixed positive integer r, we can pose a similar question concerning the rv ψ(X), when is some bounded a.e. continuous function, not necessarily admitting a pdf.