# A Discrete New Generalized Two Parameter Lindley Distribution: Properties, Estimation and Applications

• Manal M. Salem Cairo University
• Moshira A. Ismail Cairo University
Keywords: Mixture discrete distribution, Bathtub shaped, Index of dispersion, Moments, Order statistics, Maximum likelihood estimation

### Abstract

In this paper, a discrete new generalized two parameter Lindley distribution is proposed. Discrete Lindley and Geometric distributions are sub-models of the proposed distribution. Its probability mass function exhibits different shapes including decreasing, unimodal and decreasing-increasing-decreasing. Our proposed distribution has only two-parameters and its hazard rate function can accommodate increasing, constant, decreasing and bathtub shapes. Moreover, this distribution can describe equi and over dispersed data. Several distributional properties are obtained and several reliability characteristics are derived such as cumulative distribution function, hazard rate function, second hazard rate  function, mean residual life function, reverse hazard rate function, accumulated hazard rate function and also its order statistics. In addition, the study of the shapes of the hazard rate function is provided analytically and also by plots. Estimation of the parameters is done using the maximum likelihood method. A simulation study is conducted to assess the performance of the maximum likelihood estimators. Finally, the flexibility of the model is illustrated using three real data sets.

### References

Abramowitz, M., and Stegun, I. A. (1972), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series 55. Tenth Printing.‏

Bakouch, H. S., Jazi, M. A., and Nadarajah, S. (2014), A new discrete distribution. Statistics, 48(1), 200-240.

Chakraborti, S., Jardim, F., and Epprecht, E. (2017), Higher-order moments using the survival function: The alternative expectation formula. The American Statistician, 73(2), 191-194.‏

Chakraborty, S. (2015), Generating discrete analogues of continuous probability distributions-A survey of methods and constructions. Journal of Statistical Distributions and Applications, Vol. 2, pp.6.

Chakraborty, S., and Chakravarty, D. (2012), Discrete gamma distributions: properties and parameter estimations. Communications in Statistics-Theory and Methods, 41(18), 3301-3324.‏

Ekhosuehi, N., Opone, F., and Odobaire, F. (2018), A new generalized two parameter Lindley distribution. Journal of Data Science, 16(3), 549, 566.‏

El-Morshedy, M., Eliwa, M. S., and Nagy, H. (2019), A new two-parameter exponentiated discrete Lindley distribution: properties, estimation and applications. Journal of Applied Statistics, 47(2), 354-375.‏

Ghitany, M. E., Atieh, B., and Nadarajah, S. (2008), Lindley distribution and its application. Mathematics and Computers in Simulation, 78(4), 493-506.‏

Glaser, R. E. (1980), Bathtub and related failure rate characterizations. Journal of the American Statistical Association, 75(371), 667-672.‏

Gómez-Déniz, E., and Calderín-Ojeda, E. (2011), The discrete Lindley distribution: properties and applications. Journal of Statistical Computation and Simulation, 81, 1405-1416.‏

Hand, D. J., Daly, F., McConway, K., Lunn, D., and Ostrowski, E. (1993), A handbook of small data sets. cRc Press.‏

Jayakumar, K., and Babu, M. G. (2019), Discrete additive Weibull geometric distribution. Journal of Statistical Theory and Applications, 18(1), 33-45.‏

Kemp, A. W. (2004), Classes of discrete lifetime distributions. Communications in Statistics-Theory and Methods, 33(12):3069–3093.

Lindley, D. V. (1958), Fiducial distributions and Bayes' theorem. Journal of the Royal Statistical Society. Series B (Methodological), 102-107.‏

Nakagawa, T., and Osaki, S. (1975), The discrete Weibull distribution. IEEE Transactions on Reliability, Vol. 24, pp.300-301.‏

Nekoukhou, V., and Bidram, H. (2015), The exponentiated discrete Weibull distribution. Sort, 39, 127-146.‏

Nooghabi, M. S., Borzadaran, G. R. M., and Roknabadi, A. H. R. (2011), Discrete modified Weibull distribution. Metron, 69(2), 207-222.‏

Noughabi, M. S., Rezaei Roknabadi, A. H., and Mohtashami Borzadaran, G. R. (2013), Some discrete lifetime distributions with bathtub-shaped hazard rate functions. Quality Engineering, 25(3), 225-236.‏

Opone, F. C., Izekor, E. A., Akata, I. U., and Osagiede, F. E. (2021), A Discrete Analogue of the Continuous Marshall-Olkin Weibull Distribution with Application to Count Data. Earthline Journal of Mathematical Sciences, 5(2), 415-428.‏

Roy, D. (1993), Reliability measures in the discrete bivariate set-up and related characterization results for a bivariate geometric distribution. Journal of Multivariate Analysis, 46(2), 362-373.‏

Song, P., and Wang, S. (2019), A further remark on the alternative expectation formula. Communications in Statistics-Theory and Methods, 1-6.‏

Nadarajah, S., Bakouch, H. S., and Tahmasbi, R. (2012), A generalized Lindley distribution. Sankhya B, 73(2), 331-359.‏

Published
2023-08-06
How to Cite
Salem, M. M., & Ismail, M. A. (2023). A Discrete New Generalized Two Parameter Lindley Distribution: Properties, Estimation and Applications. Statistics, Optimization & Information Computing, 11(4), 978-1000. https://doi.org/10.19139/soic-2310-5070-1392
Issue
Section
Research Articles