SUBREGION-ADAPTIVE INTEGRATION OF FUNCTIONS HAVING A DOMINANT PEAK

Many statistical multiple integration problems involve integrands that have a dominant peak. In applying numerical methods to solve these pr oblems, statisticians have paid relatively little attention to existin g quadrature methods and available software developed in the numerical analysis literature. One reason these methods have been largely overl ooked, even though they are known to be more efficient than Monte Carl o for well-behaved problems of low dimensionality, may be that when ap plied naively they are poorly suited for peaked-integrand problems. In this article we use transformations based on ''split t'' distribution s to allow the integrals to be efficiently computed using a subregion- adaptive numerical integration algorithm. Our split t distributions ar e modifications of those suggested by Geweke and may also be used to d efine Monte Carlo importance functions. We then compare our approach t o Monte Carlo. In the several examples we examine here, we find subreg ion-adaptive integration to be substantially more efficient than impor tance sampling.