Available at: https://digitalcommons.calpoly.edu/theses/3307
Date of Award
6-2026
Degree Name
MS in Statistics
Department/Program
Statistics
College
College of Science and Mathematics
Advisor
Bret Holladay
Advisor Department
Statistics
Advisor College
College of Science and Mathematics
Abstract
We study precise interval estimation for the three parameters of two discrete dis- tributions: the mean parameter (i) for a Poisson distribution, the total number of successes parameter (ii), and the population size parameter (iii) of a hypergeometric distribution. In recent years, there has been development of a substantially more precise method for the estimation of parameters for the negative binomial distribution by Schilling, Doi, and Holladay (2024), the Conditional Minimal Cardinality (CMC) procedure. Following this notable achievement of improved estimation of parameters for discrete distributions, we apply the CMC method to the Poisson and hypergeometric distributions. This thesis is the first exploration of CMC procedures for the Poisson and hypergeometric distributions. To determine which existing confidence procedure has the most precise confidence intervals, we conduct comparative analyses with a focus on maximized interval precision. We compare the CMC procedures to previous well-established confidence methods in order to determine the best-performing confidence procedure for the Poisson and hypergeometric cases. In the Poisson case, we find the CMC method to be competitive in minimizing interval lengths to the method by Crow and Gardner (1959), which previously has never been surpassed by another confidence procedure. In the hypergeometric case to estimate total number of successes in a population, we find the Length/Coverage Optimal (LCO) method by Schilling and Stanley (2020) to produce the shortest intervals compared to current strict procedures. The LCO procedure in the estimation of population size appears to produce the shortest intervals among current strict procedures as well. Furthermore, we provide links to a Shiny web app for Poisson interval methods and online access to R code of both Poisson and hypergeometric CMC procedures.