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- Title
Use of the global implicit function theorem to induce singular conditional distributions on surfaces in n dimensions: Part IV.
- Authors
Bamber, Donald; Goodman, I. R.; Gupta, Arjun K.; Nguyen, Hung T.
- 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) 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 ψψ( X), when is some bounded a.e. continuous function, not necessarily admitting a pdf.
- Subjects
GLOBAL analysis (Mathematics); IMPLICIT functions; MATHEMATICAL singularities; GEOMETRIC surfaces; ESTIMATION theory; MANIFOLDS (Mathematics); VECTOR analysis; JACOBIAN matrices
- Publication
Random Operators & Stochastic Equations, 2011, Vol 19, Issue 4, p327
- ISSN
0926-6364
- Publication type
Article
- DOI
10.1515/ROSE.2011.019