A note on the modes of the negative binomial distribution of order k, type I

TL;DR

This paper derives bounds and explicit formulas for the modes of the negative binomial distribution of order k, type I, providing insights into its behavior for specific parameters, especially when p equals 0.5.

Contribution

It introduces new bounds and an explicit mode formula for the negative binomial distribution of order k, type I, based on recurrence relations.

Findings

Bounds for the modes are established using recurrence relations.

Explicit mode formula derived for p=0.5.

Mode equals k when r=1 based on bounds.

Abstract

Upper and lower bounds are derived for the mode(s) of the negative binomial distribution of order k, type I, with parameters r and p, which are employed to establish an explicit formula for the mode(s) in terms of r and k when p equals 0.5. It is also shown as a direct consequence of the upper bound alone that the mode is k when r equals 1. The derivation of the bounds is based on a known recurrence relation satisfied by the probability mass function of the distribution.