Statistical inference involving binomial and negative binomial parameters

Area of Science:

• Statistics
• Probability Theory
• Statistical Inference

Background:

• Binomial parameters are typically estimated using binomial sampling.
• Testing equality of binomial parameters before and after the first success requires different sampling methods.

Purpose of the Study:

• To develop statistical tools for testing hypotheses about two binomial parameters.
• To assess the performance of these tests in small samples and determine sample size requirements.

Main Methods:

• Utilizing negative binomial sampling for the parameter before the first success and binomial sampling for the parameter after.
• Deriving statistical tests for two hypotheses: equality to a specified value and equality to an unknown value.
• Conducting simulation studies to evaluate test accuracy, power, and robustness.

Main Results:

• The derived statistical tests accurately maintain nominal Type-I error rates in small samples.
• Simulation studies determined sample size requirements for adequate statistical power.
• The tests demonstrated robustness to certain assumption violations.

Conclusions:

• The developed statistical tools are effective for comparing binomial parameters in specific sequential Bernoulli trial scenarios.
• The tests are reliable in small samples and robust to minor assumption breaches, offering practical applications in statistical inference.