Finite-sample properties and applicability of functional CLT based confidence intervals for a population mean

Abstract

We consider a Student process that is based on independent copies of a random variable X and has trajectories in the function space D[0, 1]. If X is in the domain of attraction of the normal law, a weighted version of the Student process is known to follow a functional central limit theorem (FCLT). Accordingly, appropriate functionals of such a process converge in distribution to the same functionals of a weighted Wiener process. We use such a convergence for several functionals and derive asymptotic confidence intervals (CI) for the mean of X. Based on our investigation of the finite-sample coverage probabilities and expected lengths of the obtained CI’s for different types of distributions of X, we suggest when these FCLT based CI’s may be appealing alternatives to an asymptotic CI for the mean of X that is derived from the asymptotic normality of the Student t-statistic.