Some Properties of the Positive Hyper-Poisson Distribution and its Applications

C. Satheesh Kumar, Emil Ninan Abraham

Abstract


In this paper we consider a zero-truncated form of the hyper-Poisson distribution and investigate some of its crucial properties through deriving its probability generating function, cumulative distribution function, expressions for factorial moments, mean, variance and recurrence relations for probabilities, raw moments and factorial moments. Further, the estimation of the parameters of the distribution is discussed. The distribution has been fitted to certain real life data sets to test its goodness of fit. The likelihood ratio test procedure is adopted for checking the significance of the parameters and a simulation study is performed for assessing the efficiency of the maximum likelihood estimators.


Keywords


Confluent hypergeometric function; Mixed moment estimation; Maximum likelihood estimation; Stirling numbers of the second kind; GLRT; Simulation

Full Text:

PDF (English)

References


M. ABRAMOWITZ, I. A. STEGUN (1965). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. U.S. Government Printing Office, Washington, D.C.

M. AHMAD (2007). A short note on Conway-Maxwell-hyper Poisson distribution. Pakistan Journal of Statistics, 23, no. 2, p. 135.

G. E. BARDWELL, E. L. CROW (1964). A two-parameter family of hyper-Poisson distributions. Journal of the American Statistical Association, 59, no. 305, pp. 133–141.

E. CROW, G. BARDWELL (1965). Estimation of the parameters of the hyper-Poisson distributions. In G. P. PATIL (ed.), Classical and Contagious Discrete Distributions, Pergamon Press, New York, pp. 127–140.

P. GARMAN (1923). The European Red Mite in Connecticut Apple Orchards. Connecticut Agricultural Experiment Station.

M. HEASMAN, D. REID (1961). Theory and observation in family epidemics of the common cold. British Journal of Preventive & Social Medicine, 15, no. 1, p. 12.

P. JANI, S. SHAH (1979). On fitting of the generalized logarithmic series distribution. Journal of the Indian Society for Agricultural Statistics, 30, no. 3, pp. 1–10.

C. D. KEMP (2002). q-analogues of the hyper-Poisson distribution. Journal of Statistical Planning and Inference, 101, no. 1-2, pp. 179–183.

C. S. KUMAR (2009). Some properties of Kemp family of distributions. Statistica, 69, no. 4, pp. 311–316.

C. S. KUMAR, B. U. NAIR (2011). A modified version of hyper-Poisson distribution and its applications. Journal of Statistics and Applications, 6, no. 1/2, p. 23.

A. M. MATHAI, H. J. HAUBOLD (2008). Special Functions for Applied Scientists, vol. 4. Springer, New York, NY.

J. NEYMAN (1939). On a new class of "contagious" distributions, applicable in entomology and bacteriology. The Annals of Mathematical Statistics, 10, no. 1, pp. 35–57.

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

J. RIORDAN (1968). Combinatorial Identities. Wiley, Chichester.

A. ROOHI, M. AHMAD (2003). Estimation of the parameter of hyper-Poisson distribution using negative moments. Pakistan Journal of Statistics, 19, no. 1, pp. 99–105.

S. SINGH, K. YADAVA (1981). Trends in rural out-migration at household level. Rural Demography, 8, no. 1, p. 53.

P. J. STAFF (1964). The displaced Poisson distribution. Australian Journal of Statistics, 6, no. 1, pp. 12–20.




DOI: 10.6092/issn.1973-2201/8658