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We propose a new algorithm to compute numerically the distribution function of the sum of d dependent, non-negative random variables with given joint distribution. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2011, Vol. 17, No. 2, 562-591. This reprint differs from the original in pagination and typographic detail.

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This paper points out mistakes in some results given in the paper "Bayesian Copulae Distributions, with Application to Operational Risk Management" by Luciana Dalla Valle, published in 2009 in volume 11, number 1 of "Methodology and Computing in Applied Probability". In particular, we explain why the inverse Wishart distribution is not a conjugate prior to the Gaussian copula.

Density estimation is a central topic in statistics and a fundamental task of machine learning. In this paper, 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 has been used to measure the uniformity of the sample data points lying in each partition element. Since the resulting density estimator requires significantly less memory to be stored, it can be used in a situation where the information contained in a multivariate sample needs to be preserved, transferred or analysed.