Exchangeable Bernoulli distributions: high dimensional simulation, estimate and testing

Abstract

We explore the class of exchangeable Bernoulli distributions building on their geometrical structure. Exchangeable Bernoulli probability mass functions are points in a convex polytope and we have found analytical expressions for their extremal generators. The geometrical structure turns out to be crucial to simulate high dimensional and negatively correlated binary data. Furthermore, for a wide class of statistical indices and measures of a probability mass function we are able to find not only their sharp bounds in the class, but also their distribution across the class. Estimate and testing are also addressed.

High dimensional Bernoulli distributions: algebraic representation and applications

“…For any φ the extremal values for E[φ(S)] are reached on the extremal points (see Fontana and Semeraro (2021)). Thus, the bounds for the convex order are reached on the extremal points.…”

“…High dimensional simulation and testing is possible for some classes and under some conditions, for example see Qaqish (2003), Kang and Jung (2001), Shults (2016), Emrich and Piedmonte (1991). High dimensional simulation for exchangeable Bernoulli pmfs is addressed in Fontana and Semeraro (2021). Exchangeable Bernoulli pmfs are points in a convex polytope whose extremal points are analytically known (Fontana et al (2020)) and high dimensional simulations is possible because we know how to sample from a polytope Fontana and Semeraro (2021).…”

“…High dimensional simulation for exchangeable Bernoulli pmfs is addressed in Fontana and Semeraro (2021). Exchangeable Bernoulli pmfs are points in a convex polytope whose extremal points are analytically known (Fontana et al (2020)) and high dimensional simulations is possible because we know how to sample from a polytope Fontana and Semeraro (2021). To use this approach extremal points are necessary.…”