Simple Buehler-optimal confidence intervals on the average success probability of independent Bernoulli trials

TL;DR

This paper introduces simple, tight, Buehler-optimal one-sided confidence intervals for the average success probability of independent Bernoulli trials, applicable to non-identical parameters and sequential sampling.

Contribution

It provides analytical, tight confidence intervals for non-identical Bernoulli parameters with a guarantee of Buehler optimality across all confidence levels.

Findings

Intervals are expressed as functions of total wins, rounds, and confidence level.

Proves tightness of the bounds in the Buehler sense for all confidence levels.

Includes an application to sequential sampling scenarios.

Abstract

One-sided confidence intervals are presented for the average of non-identical Bernoulli parameters. These confidence intervals are expressed as analytical functions of the total number of Bernoulli games won, the number of rounds and the confidence level. Tightness of these bounds in the sense of Buehler, i.e. as the strictest possible monotonic intervals, is demonstrated for all confidence levels. A simple interval valid for all confidence levels is also provided with a tightness guarantee. Finally, an application of the proposed confidence intervals to sequential sampling is discussed.