Simulation-assisted saddlepoint approximation

A general saddlepoint/Monte Carlo method to approximate (conditional) multivariate probabilities is presented. This method requires a tractable joint moment generating function (m.g.f.), but does not require a tractable distribution or density. The method is easy to program and has a third-order accuracy with respect to increasing sample size in contrast to standard asymptotic approximations which are typically only accurate to the first order. The method is most easily described in the context of a continuous regular exponential family. Here, inferences can be formulated as probabilities with respect to the joint density of the sufficient statistics or the conditional density of some sufficient statistics given the others. Analytical expressions for these densities are not generally available, and it is often not possible to simulate exactly from the conditional distributions to obtain a direct Monte Carlo approximation of the required integral. A solution to the first of these problems is to replace the intractable density by a highly accurate saddlepoint approximation. The second problem can be addressed via importance sampling, that is, an indirect Monte Carlo approximation involving simulation from a crude approximation to the true density. Asymptotic normality of the sufficient statistics suggests an obvious candidate for an importance distribution. The more general problem considers the computation of a joint probability for a subvector of random T, given its complementary subvector, when its distribution is intractable, but its joint m.g.f. is computable. For such settings, the distribution may be tilted, maintaining T as the sufficient statistic. Within this tilted family, the computation of such multivariate probabilities proceeds as described for the exponential family setting.