A variational approach to a cumulative distribution function estimation problem under stochastic ambiguity

TL;DR

This paper introduces a variational method for estimating a cumulative distribution function within an ambiguity set, utilizing epi-splines and hypo-distance bounds, with proven convergence and practical algorithms demonstrated through numerical examples.

Contribution

It presents a novel variational approach using epi-splines and hypo-distance for cdf estimation under ambiguity, with convergence guarantees and an implementable linear programming algorithm.

Findings

Convergence of the approximation scheme is theoretically guaranteed.

The method effectively estimates cdfs in bivariate cases with numerical validation.

The approach integrates soft information and ambiguity considerations into cdf estimation.

Abstract

We propose a method for finding a cumulative distribution function (cdf) that minimizes the distance to a given cdf, while belonging to an ambiguity set constructed relative to another cdf and, possibly, incorporating soft information. Our method embeds the family of cdfs onto the space of upper semicontinuous functions endowed with the hypo-distance. In this setting, we present an approximation scheme based on epi-splines, defined as piecewise polynomial functions, and use bounds for estimating the hypo-distance. Under appropriate hypotheses, we guarantee that the cluster points corresponding to the sequence of minimizers of the resulting approximating problems are solutions to a limiting problem. We describe a large class of functions that satisfy these hypotheses. The approximating method produces a linear-programming-based approximation scheme, enabling us to develop an algorithm from off-the-shelf solvers. The convergence of our proposed approximation is illustrated by numerical examples for the bivariate case.