A joint central limit theorem for the sample mean and regenerative variance estimator

Abstract

Let {V(k) :K⩾1} be a sequence of independent, identically distributed random vectors in ℝd with mean vector μ. The mappingg is a twice differentiable mapping from ℝd to ℝ1. Setr=g(μ). A bivariate central limit theorem is proved involving a point estimator forr and the asymptotic variance of this point estimate. This result can be applied immediately to the ratio estimation problem that arises in regenerative simulation. Numerical examples show that the variance of the regenerative variance estimator is not necessarily minimized by using the “return state” with the smallest expected cycle length.