Main Article Content
The analysis and modeling of zero truncated count data is of primary interest in many elds such as engineering, public health, sociology, psychology, epidemiology. Therefore, in this article we have proposed a new and simple structure model, named a zero truncated discrete Lindley distribution. The
distribution contains some submodels and represents a two-component mixture of a zero truncated geometric distribution and a zero truncated negative binomial distribution with certain parameters. Several properties of the distribution are obtained such as mean residual life function, probability generating function, factorial moments, negative moments, moments of residual life function, Bonferroni and Lorenz curves, estimation of parameters, Shannon and Renyi entropies, order statistics with the asymptotic distribution of their extremes and range, a characterization, stochastic ordering and stress-strength parameter. Moreover, the collective risk model is discussed by considering the
proposed distribution as primary distribution and exponential and Erlang distributions as secondary ones. Test and evaluation statistics as well as three real data applications are considered to assess the peformance of the distribution among the most frequently zero truncated discrete probability models.
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
- Abouammoh, A.M. and Mashhour, A.F. (1981). A Note on the Unimodality of Discrete Distributions. Communication in Statistics Theory and Methods, 10(13), 1345-1354.
- Asgharzadeh, A., Bakouch, H.S., Nadarajah, S. and Sharaﬁ, F. ( 2016). A new weighted Lindley distribution with application. Brazilian Journal of Probability and Statistics, 30(1), 1-27.
- Bakouch, H. S., Al-Zahrani B. M., Al-Shomrani A. A., Marchi V. A., and Louzada F.,(2012). An extended Lindley Distribution. Journal of the Korean Statistical Society, 41, 75-85.
- Bakouch, H. S.,Jazi, M.A. and Nadarajah, S. (2014). A new discrete distribution. A Journal of Theoretical and Applied Statistics 48, 200-240.
- Bakouch, H.S. and Severini, T.A. (2009).Non Parametric Estimation in Random Sum Models. Statistica LXIX (1), 73-88.
- Bhati, D., Satry, D.V.S. and Qadri, P. Z. M. (2015).A New Generalized Poisson Lindley Distribution: Applications and Properties. Austrian Journal of Statistics, 44, 35-51.
- Borah, M. and Saikia, K.R. (2017). Zero- Truncated Discrete Shanker Distribution and Its Applications . Biometrics Biostatistics International Journal, 5(6): 00152. DOI: 10.15406/bbij.2017.05.00152.
- Brass, W.(1959). Simpliﬁed methods of ﬁtting the truncated negative binomial distribution. Biometrika 45,59-68.
- Creel, M. D. and Loomis, J. B. (1990). Theoretical and Empirical Advantages of Truncated Count Data Estimators for Analysis of Deer Hunting in California. American Journal of Agricultural Economics, 72, 434-441.
- Chakraborty, S. and Ong, S. H. (2016). A COM-Poisson type generalization of the negative binomial distribution. Communication in StatisticsTheory and Methods, 45, 4117-4135.
- Chambers, J. M., Cleveland, W. S., Kleiner, B. and Tukey, P. A. (1983). Graphical Methods for Data Analysis, Chapman and Hall,London.
- Finney, D.J. and Varley, G.C. (1955). An example of the truncated Poisson Distribution. Biometrics 11,387-391.
- Ghitany, M.E., Al-Mutairi, D. K., Balakrishnan, N. and Al-Enezi, L.J. (2013). Power Lindley distribution and associated inference. Computational Statistics and Data Analysis, 64, 20-33.
- Ghitany, M.E., Al-Mutairi, D.K. and Nadarajah, S. (2008). Zero-truncated Poisson-Lindley distribution an its application. Mathematics and Computers in Simulation 79, 279-287.
- 15. Gradshteyn, I.S. and Ryzhik, I.M. (2008).Table of Integrals,series and products, 5th Edition. Academic Press Limited,London.
- Groggan, J. T.and Carson,R. T. (1991). Models for truncated counts. Journal of Applied Econometrics, 6, 225-238.
- Hussain, T., Aslam, M. and Ahmad, M. (2016). A two parameter discrete Lindley distribution. Revista Colombiana de Estadistica, 39(1), 45-61.
- Kennedy, B.S. (2005). Does Race Predict Stroke Readmission? An Analysis Using the Truncated Negative Binomial Model. Journal of the National Medical Association, 97(5), 699-713.
- Kokonendji, C. C. and Mizre, D. (2005). Overdispersion and Underdispersion Characterization of Weighted Poisson Distribution (Technical Report No.0523). France: LMA.
- Kundu, D. and Gupta, R.D.(2009). Bivariate generalized exponential distribution. Journal of Multivariate Analysis, 100, 581-593.
- Lee, A. H., Wang, K., Yau, K. K. W and Somerford, P.G. (2003). Truncated Negative Binomial Mixed Regression Modelling of Ischaemic Stroke Hospitalizations. Statistics in Medicine, 22, 1129-1139.
- Lindley, D.V. (1958). Fiducial distributions and Bayes theorem. Journal of the Royal Statistical Society, Series B, 20, 102-107.
- Lindsay, J.K.(1995).Modeling Frequency and Count Data, Clarendon Press, Oxford University New York.
- Mir, K. A. and Ahmad, M.(2009). Size Biased Distributions and their Applications. Pakistan Journal of Statistics, 25(3), 283-294.
- Nadarajah, S., Bakouch, H. S. and Tahmasbi, R. (2011). A Generalized Lindley Distribution. Sankhya B, 73,331-359.
- Nekoukhou, V., Alamatsaz, M. H. and Bidram, H. (2012). A Discrete Analog of the Generalized Exponential Distribution. Communication in Statistics Theory and Methods, 41(11), 2000-2013.
- Phang, A. L. and Loh, E.F. (2013).Zero Truncated Strict Arcsine Model. International Journal of Computer, Electrical, Automation, Control and Information Engineering, 7(7), 989-991.
- Shanker, R. (2017). A Zero-Truncated Poisson-Amarendra Distribution and Its Application. International Journal of Probability and Statistics, 6(4),82-92
- Shanker, R. and Shukla, K.K. (2017). Zero-Truncated Poisson-Garima Distribution and its Applications. Biostat Biometrics Open Access Journal, 3(1): 555605. DOI: 10.19080/BBOAJ.2017.03.55560502.
- Shanker, R., Sharma, S. and Shanker, R. (2013). A two-parameter Lindley distribution for modeling waiting and survival times data. Applied Mathematics, 4, 363-368.
- Steutel, F. W. and Van Harn, K. (1979). Discrete Analogues of SelfDecomposability and Stability. Ann. Prob., 7, 893-899.