Logarithmically efficient estimation of the tail of the multivariate normal distribution

摘要

Simulation from the tail of the multivariate normal density has numerous applications in statistics and operations research. Unfortunately, there is no simple formula for the cumulative distribution function of the multivariate normal law, and simulation from its tail can frequently only be approximate. In this article we present an asymptotically efficient Monte Carlo estimator for quantities related to the tail of the multivariate normal distribution. The estimator leverages upon known asymptotic approximations. In addition, we generalize the notion of asymptotic efficiency of Monte Carlo estimators of rare-event probabilities to the sampling properties of Markov chain Monte Carlo algorithms. Regarding these new notions, we propose a simple and practical Markov chain sampler for the normal tail that is asymptotically optimal. We then give a numerical example from finance that illustrates the benefits of an asymptotically efficient Markov chain Monte Carlo sampler.