Targeting Kollo skewness with random orthogonal matrix simulation

High-dimensional multivariate systems often lack closed-form solutions and are therefore resolved using simulation. Random Orthogonal Matrix (ROM) simulation is a state-of-the-art method that has gained popularity in this context because certain simulation errors are completely removed (Ledermann et al., 2011). Specifically, every sample generated with ROM simulation exactly matches a target mean and covariance matrix. A simple extension can also target a scalar measure of skewness or kurtosis. However, targeting non-scalar measures of higher moments is much more complex. The problem of targetting exact Kollo skewness has already been considered, but the algorithm proceeds via time-consuming trial-and-error which can be very slow. Moreover, the algorithm often fails completely. Furthermore, it produces simulations with very long periods of inactivity which are inappropriate for most real-world applications. This paper provides an in-depth theoretical analysis of a much quicker ROM simulation extension which always succeeds to target Kollo skewness exactly. It also derives new results on Kollo skewness in concatenated samples and applies them to produce realistic simulations for many applications, especially those in finance. Our first contribution is to establish necessary and sufficient conditions for Kollo skewness targeting to be possible. Then we introduce two novel methods, one for speeding up the algorithm and another for improving the statistical properties of the simulated data. We illustrate several new theoretical results with some extensive numerical analysis and we apply the algorithm using some real data drawn from two different multivariate systems of financial returns.