On the modes of the negative binomial distribution of order k, type I

Abstract

Upper and lower bounds are derived for the modes of the negative binomial distribution of order k, type I, with parameter vector (r, p). The bounds are employed to establish an explicit formula for the modes in terms of k and r when p equals to 0.5. It is also shown that the mode is k when r equals 1. For computational purposes, two efficient algorithms are developed. The first evaluates the modes and the second simulates values of the distribution. An application connected to continuous sampling plans is discussed. Numerical examples illustrate further the results of the article.

Keywords:

• Bounds
• Binary trials
• Discrete distributions
• Modes
• Negative binomial distribution
• Runs

Mathematics Subject Classification:

• 60E05
• 62E15
• 62E20
• 62P30

Acknowledgments

The authors would like to thank the anonymous referee for the thorough reading and useful comments and suggestions which helped to improve the paper.