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

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, say , ( , ), which are employed to establish an explicit formula for ( , ) in terms of r and k when p = 1/2. It is also shown as a direct consequence of the upper bound alone that ( , ) = when r = 1. The derivation of the bounds is based on a known recurrence relation satisfied by the probability mass function of , ( , ).

The Modes of the Poisson Distribution of Order 3 and 4

“…See, e.g., Ling [ 8 ], Mohanty [ 9 ], Chang [ 10 ], Johnson et al [ 11 ], Shmueli and Kohen [ 12 ], Balakrishnan and Koutras [ 13 ], Fu and Lou [ 14 ], Eryilmaz [ 15 ], Rakitzis and Antzoulakos [ 16 ], Dafnis et al [ 17 ], Sengar et al [ 18 ], Kwon [ 19 ], and references therein. However, the modes of these distributions are not yet known, except for the modes of the geometric distribution of order k and partial results for the mode(s) of the Poisson distribution of order k and the negative binomial distribution of the same order derived by Luo [ 20 ], Georghiou et al [ 21 ], Philippou [ 22 ], Shao and Fu [ 23 ], and Georghiou et al [ 24 ].…”