Using Stein’s Equation in Simulation

Stein’s Poisson equation was developed to enable one to bound the error in approximating the distribution of the sum of Bernoulli random variables by a Poisson distribution. In this talk we explain how Stein’s equation is derived and then show how it can be utilized in simulation. In particular we present evidence that our approach based on the Stein equation results in simulation estimators with extremely small variances when compared with other approaches. We will indicate this both analytically and empirically. We will discuss applications to such problems as determining the number of times that successive partial sums of random variables fall outside of defined limits; analyzing generalized birthday and coupon collecting problems; determining the probability that a reliability system functions, and determining the distribution of the time until a pattern appears in Markov chain generated data