A three parameter hyper-Poisson distribution and some of its properties


  • C. Satheesh Kumar University of Kerala
  • B. Unnikrishnan Nair M. S. M. College, Kayamkulam




Confluent hypergeometric series, Displaced Poisson distribution,


A new class of distribution is introduced here as a generalization of the well-known hyper-Poisson distribution of Bardwell and Crow (J. Amer. Statist. Associ., 1964) and alternative hyper-Poisson distribution of Kumar and Nair (Statistica, 2012), and derive some of its important aspects such as mean, variance, expressions for its raw moments, factorial moments, probability generating function and recursion formulae for its probabilities, raw moments and factorial moments. The estimation of the parameters of the distribution by various methods are considered and illustrated using some real life data sets. Further, a test procedure is suggested for testing the significance of the additional parameter and a simulation study is carried out for comparing the performance of the estimators.


M. ABRAMOWITZ, I.A. STEGUN (1965). Hand book of Mathematical Functions. Dover, New York.

M. AHMAD (2007). A short note on Conway-Maxwell-hyper Poisson distribution. Pakistan Journal of Statistics , 23, pp. 135-137.

P.S. ALBERT (1991). A two-state Markov mixture model for a time series of epileptic seizure Counts. Biometrics, 47, pp. 1371-1381.

G.E. BARDWELL, E.L. CROW (1964). A two parameter family of hyper-Poisson distributions. Journal of American statistical Association, 59, pp. 133-141.

E.L. CROW, G.E. BARDWELL. (1965). Estimation of the parameters of the hyper-Poisson distributions. Classical and Contagious Discrete Distributions.

G. P. Patil (editor), pp. 127-140, Pergamon Press, Oxford.

P. GARMAN (1951). Original data on European red mite on apple leaves. Connecticut.

D.J.HAND, F. DALY, A.D. LUNN, K.J. MCCONWAY ,E. OSTROWSKI (1994). A Hand Book of Small Data Sets. Chapman and Hall, London.

N.L. JOHNSON, A.W. KEMP, S.KOTZ (2005). Univariate Discrete Distributions. Wiley, New York.

C.D KEMP, A.W. KEMP (1965). Some properties of the Hermite distribution. Biometrika, 52, pp. 381-394.

C.D. KEMP (2002). q-analogues of the hyper-Poisson distribution. Journal of Statistical Planning and Inference, 101, pp. 179-183.

C.S. KUMAR, B.U. NAIR (2011). Modi_ed version of hyper-Poisson distribution and its applications. Journal of Statistics and Applications, 6, pp. 23-34.

C.S. KUMAR, B.U. NAIR (2012a). An extended version of hyper-Poisson distribution and some of its applications. Journal of Applied Statistical Sciences, 19, pp. 81-88.

C.S. KUMAR, B.U. NAIR (2012b). An alternative hyper-Poisson distribution. ‘Statistica, 72(3), pp. 357-369.

A.M. MATHAI, H.J. HAUBOLD (2008). Special Functions for Applied Scientists. Springer, New York.

T. NISIDA (1962). On the multiple exponential channel queuing system with hyper-Poisson arrivals. Journal of the Operations Research Society, 5, pp. 57-66.

C.R. RAO (1973). Linear Statistical Inference and its Applications (second edition). Johnwiley, New York.

J. RIORDAN (1968). Combintorial identities. Wiley, New York.

A. ROOHI, M. AHMAD (2003a). Estimation of the parameter of hyper-Poisson distribution using negative moments. Pakistan Journal of Statistics, 19, pp. 99-105.

A. ROOHI, M. AHMAD (2003b). Inverse ascending factorial moments of the hyper-Poisson probability distribution. Pakistan Journal of Statistics, 19, pp. 273-280.

P.J. STAFF (1964). The displaced Poisson distribution. Australian Journal of Statistics, 6, pp. 12-20.

G.M. STIRRETT, G. BEALL, M. TIMONIOA (1937). A eld experiment on the control of the European corn borer. Pyrausta nubilalis Hubn. by Beauveria bassiana Vuill.Scient. Agri 17, pp. 587-591.




How to Cite

Kumar, C. S., & Nair, B. U. (2014). A three parameter hyper-Poisson distribution and some of its properties. Statistica, 74(2), 183–198. https://doi.org/10.6092/issn.1973-2201/5000