Simulation in Exponential Families

An acceptance-rejection algorithm for the simulation of random variables in statistical exponential families is described. This algorithm does not require any prior knowledge of the family, except sufficient statistics and the value of the parameter. It allows simultaneous simulatation from many members of the exponential family. We present some bounds on computing time, as well as the main properties of the empirical measures of samples simulated by our methods (functional Glivenko-Cantelli and central limit theorems). This algorithm is applied in order to evaluate the distribution of M-estimators under composite alternatives; we also propose its use in Bayesian statistics in order to simulate from posterior distributions.