Improved Methods for Exact Confidence Intervals When Sampling With and Without Replacement for Rare Events
Abstract
This paper addresses the problem of estimating the probability or the count of successes observed when taking Bernoulli samples from finite or infinite populations. Exact confidence intervals also have expected lengths (or widths) that are the same or even smaller than those of the approximations. Fractional bounds are introduced for the Hypergeometric and Negative Hypergeometric Distributions, which achieve minimum coverage probabilities closer to the desired levels than those achievable using the customary integer solutions. Spreadsheet formulas and macros are given that provide one-line solutions for Exact bounds. Point estimates of success probabilities or success counts are also addressed. Expressions by Haldane for the Negative Binomial Distribution and Guenther for the Negative Hypergeometric Distribution are demonstrated to be superior to naive point estimates. An improved point estimate for the Negative Binomial Distribution ending with a single success (the Geometric Distribution) is proposed and tested.
Document Details
- Document Type
- Technical Report
- Publication Date
- Aug 17, 2023
- Accession Number
- AD1208335
Entities
People
- Warren H. Jr Debany