It has been shown that a sum of independent observations from a logarithmic series distribution gives a negative binomial distribution when the number of terms in the sum has a Poisson distribution.
I would like the earliest reference to an equivalent way of looking at this, which I think is more intuitive (I assume I wasn't the first to think of it). Suppose that events involving k individuals have a Poisson distribution with rate proportional to p^k / k. If these events are independent then the total number of individuals involved has a negative binomial distribution.
Hello,
We have studied the negative binomial process in several papers. In the following paper, that is also available through Research Gate you will find an extensive review of the literature, where this cluster effect has been pointed out. Hope this will help
Article: Distributional properties of the negative binomial Lévy process
Tomasz J. Kozubowski, Krzystof Podgórski
Probability and Mathematical Statistics 01/2009; 29(1).
https://www.researchgate.net/publication/265549266_Distributional_properties_of_the_negative_binomial_Lvy_process
Some related information can be also find in:
Article: Invariance properties of the negative binomial Levy process and stochastic self-similarity
Tomasz J Kozubowski, Krzysztof Podgórski
International Mathematical Forum. 01/2007; 2(30):1457-1468.
Article Distributional properties of the negative binomial Lévy process
Thank you. I knew about some of the references, e.g. Greenwood and Yule (1920) that introduced the gamma weighted mixture of Poissons, and Fisher (1941) that dealt with a sum of a Poisson distruted number of terms of logarithmic series variables. The earlier references are very helful.
- Badges
- Science topic
- Similar topics
- Mathematics
- Statistics
- Applied Statistics
- Survey Design
- Questionnaire