Exact Optimal Fixed Width Confidence Interval Estimation for the Mean

VIKAS DEEP, Achal Bassamboo, Sandeep Kumar Juneja, assaf zeevi

Abstract: We consider a classical problem in simulation/statistics - given i.i.d. samples of a rv, the goal is to arrive at a confidence interval (CI) of a pre-specified width ε , and with a coverage guarantee that the mean lies in the CI with probability at least 1−δ for pre-specified δ∈(0,1) . This problem has been well studied in an asymptotic regime as ε shrinks to zero. The novelty of our analysis is the derivation of the lower bound on the number of samples required by any algorithm to construct a CI of ε –width with the coverage guarantee for fixed ε>0 and δ , and construction of an algorithm that, under mild assumptions, matches the lower bound. For simplicity, we present our results for rv belonging to a single parameter exponential family, and illustrate its efficacy through a numerical study.