Teachable normal approximations to binomial and related probabilities or confidence bounds
Lutz Mattner
TL;DR
This work develops teachable, finite-sample normal-approximation bounds for binomial-related distributions (hypergeometric, binomial, Poisson) to yield interpretable confidence bounds such as Clopper–Pearson. It unifies interval-probability bounds under precise error terms of order $1/\sigma$ and provides refined results for symmetric binomial laws using continuity corrections, along with rigorous, practical bounds for Clopper–Pearson bounds. The methodology rests on Esseen–Shevtsova type refinements and Bernoulli-convolution theory, with extensions and sharpened constants for binomial and symmetric cases, and应用s to CP-bound approximations via contemporary inequalities. The findings enhance teachability and practical use by delivering explicit, verifiable inequalities rather than asymptotic limits, supporting students and practitioners in constructing accurate, accessible normal-approximation intervals.
Abstract
This document is an extended version of an abstract for a talk, with approximately the same title, to be held at the 7th Joint Statistical Meeting of the Deutsche Arbeitsgemeinschaft Statistik, from 24 to 28 March 2025 in Berlin. Here ``teachable'' is meant to apply to people ranging from sufficiently advanced high school pupils to university students in mathematics or statistics: For understanding most of the proposed approximation results, it should suffice to know binomial laws, their means and variances, and the standard normal distribution function (but not necessarily the concept of a corresponding normal random variable). Of the proposed approximations, some are well-known (at least to experts), and some are based on teaching experience and research at Trier University.