Normal variance mixtures: Distribution, density and parameter estimation.

Efficient computation of the distribution and log-density function of multivariate normal variance mixtures as well as a likelihood-based fitting procedure for those distributions are presented. Existing methods for the evaluation of the distribution function of a multivariate normal and t distribution are generalized to an efficient randomized quasi-Monte Carlo (RQMC) algorithm that is able to estimate the probability that a random vector following a multivariate normal variance mixture distribution falls into a (possibly unbounded) hyper rectangle, as long as the mixing variable has a tractable quantile function. The log-density is approximated using an adaptive RQMC algorithm. Parameter estimation for multivariate normal variance mixtures is achieved through an expectation-maximization-like algorithm where all weights and log-densities are numerically estimated. It is demonstrated through numerical examples that the suggested algorithms are quite fast; even for high dimensions around 1000 the distribution function can be estimated with moderate accuracy using only a few seconds of run time. Even log-densities around -100 can be estimated accurately and quickly. An implementation of all algorithms presented in this work is available in the R package nvmix (version >= 0.0.2).