Modelling multivariate systems is important for many applications in engineering and operational research. The multivariate distributions under scrutiny usually have no analytic or closed form. Therefore their modelling employs a numerical technique,
typically multivariate simulations, which can have very high dimensions. Random Orthogonal Matrix (ROM) simulation is a method that has gained some popularity because of the absence of certain simulation errors. Specifically, it exactly matches a target mean, covariance matrix and certain higher moments with every simulation. This paper extends the ROM simulation algorithm presented by Hanke et al. (2017), hereafter referred to as HPSW, which matches the target mean, covariance matrix and Kollo skewness vector exactly. Our first contribution is to establish necessary and sufficient conditions for the HPSW algorithm to work. Our second contribution is to develop a general approach for constructing admissible values in the HPSW. Our third theoretical contribution is to analyse the effect of multivariate sample concatenation on the target Kollo skewness. Finally, we illustrate the extensions we develop here using a simulation study.
Interest in predicting multivariate probability distributions is growing due to the increasing availability of rich datasets and computational developments. Scoring functions enable the comparison of forecast accuracy, and can potentially be used for
estimation. A scoring function for multivariate distributions that has gained some popularity is the energy score. This is a generalization of the continuous ranked probability score (CRPS), which is widely used for univariate distributions. A little-known, alternative generalization is the multivariate CRPS (MCRPS). We propose a theoretical framework for scoring functions for multivariate distributions, which encompasses the energy score and MCRPS, as well as the quadratic score, which has also received little attention. We demonstrate how this framework can be used to generate new scores. For univariate distributions, it is well-established that the CRPS can be expressed as the integral over a quantile score. We show that, in a similar way, scoring functions for multivariate distributions can be disintegrated to obtain scoring functions for level sets. Using this, we present scoring functions for different types of level set, including those for densities and cumulative distributions. To compute the scoring functions, we propose a simple numerical algorithm. We illustrate our proposals using simulated and stock returns data.
