Functional asymptotic confidence intervals for a common mean of independent random variables

TL;DR

This paper develops data-based functional central limit theorems for independent random variables with a common mean, enabling the construction of new, practical asymptotic confidence intervals for the mean.

Contribution

It establishes fully data-driven versions of existing FCLTs, extending their use for creating functional asymptotic confidence intervals for the common mean.

Findings

Data-based FCLTs are established for common mean estimation.

New functional asymptotic confidence intervals are introduced.

Applications include two specific examples of these confidence intervals.

Abstract

We consider independent random variables (r.v.'s) with a common mean $\mu$ that either satisfy Lindeberg's condition, or are symmetric around $\mu$. Present forms of existing functional central limit theorems (FCLT's) for Studentized partial sums of such r.v.'s on $D[0,1]$ are seen to be of some use for constructing asymptotic confidence intervals, or what we call functional asymptotic confidence intervals (FACI's), for $\mu$. In this paper we establish completely data-based versions of these FCLT's and thus extend their applicability in this regard. Two special examples of new FACI's for $\mu$ are presented.