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

Abstract

This four-part 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) . 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 ( X ), when is some bounded a.e. continuous function, not necessarily admitting a pdf.