A generalization of the negative binomial distribution
Abstract
Background & Aim: Consider a sequence of independent Bernoulli trials with p denoting the probability of success at each trial. With this definition, the probability that the nth success proceed by r failures follows the negative binomial distribution (NB). NB model has been derived from two different forms. At first, the NB can be thought as a Poisson-gamma mixture. The second form of the NB can be derived as a full member of a single parameter exponential family distribution, and therefore considered as a GLM (generalized linear models).
Methods & Materials: We have described a new generalized NB (GNB) distribution with three parameters α, β and k obtained as a compound form of the generalized Poisson and gamma distributions. This distribution gives a very close fit for a large number of data and provides an appropriate model for numerous studies. The most important feature of this model is, its time dependent probabilities, and also it can be used for a variety of researches especially in the survival analysis.
Results: This model has been illustrated with two datasets that are indirect measures of illness, along comparing the results of the fitting with NB. Results indicate too much satisfaction. Expected frequencies have been calculated for these data sets to show that the distribution provides a very satisfactory fit in different situations.
Conclusion: Using GNB models allows analyzing very complex data. This distribution gives a very close fit for a large number of data and provides an appropriate model for numerous studies. With k = 0 the model becomes the ordinary NB and with α = 1, it becomes a new model which we call it the generalized geometric distribution with two parameters. The most important feature of this model is its time-dependent probabilities.
Greenwood M, Yule U. An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to the occurrence of multiple attacks of disease or of repeated accidents. Royal Statistical Society 1920; 83(2): 255-79.
Johnson NL, Kotz S. Discrete distributions. New York, NY: Wiley; 1969. p. 328.
Takacs L. A generalization of the ballot problem and its application in the theory of queues. J Am Stat Assoc 1962; 57(298): 327-37.
Mohanty SG. On a generalised two-coin tossing problem. Biom Z 1966; 8(4): 266-72.
Jain GC, Consul PC. A generalized negative binomial distribution. SIAM J Appl Math 1971; 21(4): 501.
Cameron AC, Trivedi PK. Regression analysis of count data. 1st ed. Cambridge, UK: Cambridge University Press; 1998. p. 411.
Winkelmann R, Zimmermann K. Recent Developments in Count Data Modelling
Theory and Application. J Econ Surv 1995; 9(1): 1-24.
Greene WH. LIMDEP Econometric Modeling Guide. Version 9. Plainview, NY: Econometric Software Inc; 2006.
Brass W. The distribution of births in human populations in rural Taiwan. Population Studies 1958; 12(1): 51-72.
Dandekar VM. Certain modified forms of binomial and poisson distributions. Sankhya 1955; 15(3): 237-50.
Consul PC, Jain GC. A generalization of the poisson distribution. Technometrics 1973; 15(4): 791-9.
Canada Dominion Bureau of Statistics, Canada Department of National Health and Welfare. Canadian Sickness Survey 1950-51. No.8. Volume of Health Care (National Estimates). DBS Reference Paper no.51. Ottawa, Canada: The Queen's Printer and Controller of Stationary; 1955.
Chiang CL. An index of health: mathematical models. Vital Health Stat 1 1965; (94): 1-19.
| Files | ||
| Issue | Vol 2 No 3 (2016) | |
| Section | Original Article(s) | |
| Keywords | ||
| Negative binomial Generalized Poisson Gamma distribution Compound distribution | ||
| Rights and permissions | |
|
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. |
