Robust estimation with latin hypercube sampling: a central limit theorem for Z-estimators

Latin hypercube sampling (LHS) is a widely used stratified sampling method in computer experiments. In this work, we extend the existing convergence results for the sample mean under LHS to the broader class of $Z$-estimators, estimators defined as the zeros of a sample mean function. We derive the asymptotic variance of these estimators and demonstrate that it is smaller when using LHS compared to traditional independent and identically distributed (i.i.d.) sampling. Furthermore, we establish a Central Limit Theorem for $Z$-estimators under LHS, providing a theoretical foundation for its improved efficiency.