# Weighted Negative Binomial-Poisson Lindley with Application to Genetic Data

### Abstract

Background & Aim: Mixed Poisson and mixed negative binomial distributions have been considered as alternatives for fitting count data with over-dispersion. This study introduces a new discrete distribution which is a weighted version of Poisson-Lindley distribution.Methods & Materials: The weighted distribution is obtained using the negative binomial weight function and can be fitted to count data with over-dispersion. The p.m.f., p.g.f. and simulation procedure of the new weighted distribution, namely weighted negative binomial- Poisson-Lindley (WNBPL), are provided. The maximum likelihood method for parameters estimation is also presented.Results: The WNBPL distribution is fitted to several datasets, related to genetics and compared with the Poison distribution. The goodness of fit test shows that the WNBPL can be a useful tool for modeling genetics datasets. Conclusion: This paper introduces a new weighted Poisson-Lindley distribution which is obtained using negative binomial weight function and can be used for fitting over-dispersed count data. The p.m.f., p.g.f. and simulation procedure are provided for the new weighted distribution, namely the weighted negative binomial-Poisson Lindley (WNBPL) to better inform parents from possible time of occurrence reflux and treatment strategies.### References

(1) Bulmer, M.G. 1974. On fitting the Poisson-lognormal distribution to species-abundance data. Biometrics, 101-110.

(2) Cancho, V.G., Louzada-Neto, F., and Barriga, G.D.C. 2011. The Poisson-exponential lifetime distribution. Computational Statistics & Data Analysis, 55(1): 677-686.

(3) Castillo, J. and Perez-Casany, M. 2005. Overdispersed and underdispersed Poisson generalizations. Journal of Statistical Planning and Inference, 134: 486-500.

(4) Catcheside DG, Lea DE, Thoday JM (1946) Types of chromosome structural change induced by the irradiation on Tradescantia microspores. Journal of Genetics 47: 113-136.

(5) Catcheside DG, Lea DE, Thoday JM (1946) The production of chromosome structural changes in Tradescantia microspores in relation to dosage, intensity and temperature. Journal of Genetics l.47: 137-149.

(6) Denuit, M. 1997. A new distribution of Poisson-type for the number of claims. Astin Bulletin, 27(2): 229-242.

(7) Ghitany, M.E., Atieh B. and Nadarajah, S. 2008. Lindley distribution and its application. Math. Comput. Simulat., 78: 493-506.

(8) Gomez-Deniz, E., Sarabia, J.M. and Calderin-Ojeda, E. 2008. Univariate and multivariate versions of the negative binomial-inverse Gaussian distributions with applications. Insurance: Math. Econ., 42: 39-49.

(9) Hussain, T., Aslam, M. and Ahmad, M. (2016). A two parameter discrete Lindley

Distribution. Revista Colombiana de Estadistica, 39(1): 45-61.

(10) Klugman, S., Panjer, H., Willmot, G. 2012. Loss models: from data to decision. 4th Ed. John Wiley and Sons, USA.

(11) Kokonendji, C. C. and Casany, M. P. (2012). A Note on Weighted Count Distributions. Journal of Statistical Theory and Applications, 11(4): 337-352.

(12) Lord, D. and Geedipally. S.R. 2011. The negative binomial-Lindley distribution as a tool for analyzing crash data characterized by a large amount of zeros. Accident Anal. Prevent., 43: 1738-1742.

(13) Meng, S., Wei, Y. and Whitmore, G.A. 1999. Accounting for individual overdispersion in a bonus-malus system. ASTIN Bull., 29: 327-337.

(14) Neel, J. V. and Schull, W. J. (1966). Human Heredity. University of Chicago Press, Chicago,

-227.

(15) Rama Shanker, Hagos Fesshaye. 2015. On Poisson Lindley Distribution and Applications to Biological Sciences. Biometrics & Biostatistics International Journal. 2(4)

(16) Patil, G.P., Rao, C.R. and Ratnaparkhi, M.V. 1986. On discrete weighted distribution and their use in model choice for observed data. Commun. Statis. Theory Math., 15(3): 907-918.

(17) Rao, C.R. 1965. Weighted distributions arising out of methods of ascertainment. In Classical and Contagious Discrete Distributions, G. P. Patil (Eds). Calcuta: Pergamon Press and Statistical Publishing Society, 320-332.

(18) Ridout, M.S. and Besbeas, P. 2004. An empirical model for underdispersed count data. Statistical Modelling, 4: 77-89.

(19) Sankaran, M. 1970. The discrete Poisson-Lindley distribution. Biometrics, 26: 145-149.

(20) Shanker, R. and Mishra, A. 2014. A two parameter Poisson-Lindley distribution. International Journal of Statistics and Systems, 9(1): 79-85.

(21) Shanker, R., Sharma, S., Shanker, U., Shanker, R., and Leonida, T.A. 2014. The discrete Poisson-Janardan distribution with applications. International Journal of Soft Computing and Engineering, 4(2): 31-33.

(22) Shmueli, G., Minka, T.P., Kadane, J.P., Borle, S. and Boatwright, P. 2005. A useful distribution for ftting discrete data: revival of the Conway-Maxwell-Poisson distribution. Journal of the Royal Statistical Society, Ser. C., 54: 127-142.

(23) Simon, L. 1961. Fitting negative binomial distributions by the method of maximum likelihood. Proceedings of the Casualty Actuarial Society, XLVIII: 45-53.

(24) Trembley, L. 1992. Using the poisson inverse-gaussian in bonus-malus systems. ASTIN Bulletin, 22: 97-106.

(25) Willmot, G.E. 1987. The Poisson-inverse Gaussian distribution as an alternative to the negative binomial. Scandinavian Actuarial J., 2: 113-127.

(26) Zamani, H. and Ismail, N. 2010. Negative binomial-Lindley distribution and its application. J. Math. Statist., 6: 4-9.

(27) Zamani, H., Ismail, N. and Faroughi, P. 2014. Poisson-weighted exponential: Univariate version and regression model with applications. J. Math. Statist., 10: 148-154.