Nonparametric density estimation with Wasserstein distance for actuarial applications

Density estimation is a central topic in statistics and a fundamental task of actuarial sciences. In this work, we present an algorithm for approximating multivariate empirical densities with a piecewise constant distribution defined on a hyperrectangular-shaped partition of the domain. The piecewise constant distribution is constructed through a hierarchical bisection scheme, such that locally, the sample cannot be statistically distinguished from a uniform distribution. The Wasserstein distance represents the basic element of the bisection technique, and has been used to measure the uniformity of the sample data points lying in each partition element. Since the resulting density estimator can be efficiently and concisely represented%requires significantly less to be stored, it can be used whenever the information contained in a multivariate sample needs to be preserved, transferred or analysed. Also, the proposed methodology is peculiar because Wasserstein distance makes it possible to establish an upper bound on the absolute deviation, in terms of tail value at risk, between original sample and estimator marginals. For these features, our algorithm can play an important role in nowadays insurance and financial environments characterised by greater complexity, increasingly interconnections and advanced quantitative approaches. Its applications range from pricing, to capital modelling and, in general, to all those contexts where multivariate problems arise.