Modularity Classes and Boundary Effects in Multivariate Stochastic Dominance

TL;DR

This paper extends classical multivariate stochastic dominance results to distributions with compact support, providing a direct proof that clarifies the roles of modularity and boundary effects, useful for statistical dominance testing.

Contribution

It offers a new direct proof for stochastic dominance with compact support distributions, relaxing previous independence and continuity assumptions.

Findings

Extended stochastic dominance conditions to general compact support distributions.

Clarified the role of modularity classes and boundary effects.

Provided a foundation for statistical tests without absolute continuity.

Abstract

Hadar and Russell (1974) and Levy and Paroush (1974) presented sufficient conditions for multivariate stochastic dominance when the distributions involved are continuous with compact support. Further generalizations involved either independence assumptions (Sacarsini (1988)) or the introduction of new concepts like 'correlation increasing transformation' (Epstein and Tanny (1980), Tchen (1980), Mayer (2013)). In this paper, we present a direct proof that extends the original results to the general case where the involved distributions are only assumed to have compact support. This result has in turn proven useful for statistical tests of dominance without the assumption of absolute continuity. The first section introduces several concepts used throughout the paper. In the second section we recall the classic result as presented in Atkinson and Bourguignon (1982), with a slightly lighter proof using the general integration by parts formula for n dimensional Lebesgue-Stieljes integrals. In the third section we present our proof of the general result, using Riemman-Stieljes partial sums in a direct fashion that helps to clarify the role of modularity conditions and boundary effects in the sufficiency of the conditions. The last section discusses the relevance of the result and concludes.