Copulas and Histogram-Valued Data

Abstract

Histogram-valued data are emerging increasingly often as a consequence of the aggregation of large datasets. One statistic that underpins many methodologies especially regression and principal component analyses is the covariance function. To date, no method exists for calculating these functions directly from the marginal histogram observations. This article develops techniques through copula functions to develop a parametric distribution for multivariate histogram-valued data. In particular, maximum likelihood, inference function for margins, and canonical maximum likelihood estimation methods are proposed. A numerical study helps to ascertain which copulas are best to use in various cases, and thence to calculate the covariances. The results are applied to a real dataset.