On the modes of the negative binomial distribution of order

Jiguang Shao and Sheng Fu

Journal of Applied Statistics, 2016, vol. 43, issue 11, 2131-2149

Abstract: In this paper, the modes of the negative binomial distribution of order k are studied. Firstly, the method of transition probability flow graphs is introduced to deal with the probability-generating function of the geometric distribution of order k, which is a special case of the negative binomial distribution of the same order. And then, the general negative binomial distribution of order k is investigated. By means of probability distribution function, the mode of the geometric distribution of order k is derived, i.e. $ m_{X_{(k)}}=k $ mX(k)=k. Based on the Fibonacci sequence and Poly-nacci sequence, the modes of the negative binomial distribution of order k in some cases are obtained: (1) $ m_{X_{(2,2)}}=6, 7, 8 $ mX(2,2)=6,7,8 and $ m_{X_{(3,2)}}=16 $ mX(3,2)=16, for p=0.5; (2) $ m_{X_{(2,3)}}=13 $ mX(2,3)=13 for p=0.5. Finally, an application of negative binomial distribution of order k in continuous sampling plans is given.

Date: 2016
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DOI: 10.1080/02664763.2015.1130802

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