A New Two-Parameter Discrete Weighted Lindley Distribution: Properties, Estimation and Applications

Manal  Mahmoud Salem; Moshira  Ahmed Ismail

Authors

Manal Mahmoud Salem Department of Statistics, Cairo University, Cairo, Egypt
Moshira Ahmed Ismail Department of Statistics, Cairo University, Cairo, Egypt

Keywords:

Mixture discrete distribution, discretizing using the reliability function, hazard rate function, bathtub shaped, index of dispersion

Abstract

A new two-parameter discrete weighted Lindley distribution is proposed. The proposed distribution can be seen as a generalization of one parameter discrete Lindley distribution which has only increasing hazard rate whereas our proposed distribution has increasing and bathtub shaped hazard rates. In addition, this distribution can describe over, equi and under dispersed data. Some properties of the proposed distribution are obtained such as cumulative distribution function, hazard rate function, second hazard rate function, mean residual life function, reversed hazard rate function, accumulated hazard rate function as well as order statistics. In addition, an analytical study of the shapes of the hazard rate function is provided. Maximum likelihood estimation of the parameters of the proposed distribution is considered. The results obtained in this study are illustrated using both simulated and four real data sets.

References

Abramowitz M, Stegun IA, editors. Handbook of mathematical functions with formulas, graphs, and mathematical tables. 10th printing, with corrections. Washington (DC): National Bureau of Standards; 1972. (Applied Mathematics Series; no. 55).

Alamatsaz MH, Dey S, Dey T, Harandi SS. Discrete generalized Rayleigh distribution. Pak J Stat. 2016; 32(1):1-20.

Ali S. On the Bayesian estimation of the weighted Lindley distribution. J Stat Comput Simul. 2013; 85(5): 855-880.

Almalki SJ, Nadarajah S. A new discrete modified Weibull distribution. IEEE Trans Reliab. 2014; 63(1): 68-80.

Almetwally EM, Abdo DA, Hafez EH, Jawa TM, Sayed-Ahmed N, Almongy HM. The new discrete distribution with application to COVID-19 data. Results Phys. 2022; 32: 104987. http://doi.org/

1016/j.rinp.2021.104987

Al-Mutairi DK, Ghitany ME, Kundu D. Inferences on stress-strength reliability from weighted Lindley distributions. Commun Stat Theory Methods. 2015; 44(19): 4096-4113.

Bashir S and Rasul M. Some properties of the weighted Lindley distribution. Int J Econ Bus Rev. 2015; 3(8): 11-17.

Bodhisuwan W, Sangpoom S. The discrete weighted Lindley distribution. In: 2016 12th International Conference on Mathematics, Statistics, and Their Applications; 2016 Oct 4-6; Banda Aceh, Indonesia. IEEE; 2016.

Chakraborti S, Jardim F, Epprecht E. Higher-order moments using the survival function: the alternative expectation formula. Am Stat. 2017; 73(2): 191-194.

Chakraborty S. Generating discrete analogues of continuous probability distributions: a survey of methods and constructions. J Stat Distrib Appl. 2015; 2: 6. http://doi.org/10.1186/s40488-015-0028-6

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

El-Morshedy M, Eliwa MS, Nagy H. A new two-parameter exponentiated discrete Lindley distribution: properties, estimation and applications. J Appl Stat. 2019; 47(2): 354-375.

Ghitany ME, Alqallaf F, Al-Mutairi DK, Husain HA. A two-parameter weighted Lindley distribution and its applications to survival data. Math Comput Simul. 2011; 81(6): 1190-1201.

Ghitany ME, Atieh B, Nadarajah S. Lindley distribution and its application. Math Comput Simul. 2008; 78(4): 493-506.

Ghitany ME, Song P, Wang S. New modified moment estimators for the two-parameter weighted Lindley distribution. J Stat Comput Simul. 2017; 87(16): 3225-3240.

Glaser RE. Bathtub and related failure rate characterizations. J Am Stat Assoc. 1980; 75(371): 667-672.

Gómez-Déniz E and Calderín-Ojeda E. The discrete Lindley distribution: properties and applications. J Stat Comput Simul. 2011; 81(11): 1405-1416.

Jayakumar K, Babu MG. Discrete additive Weibull geometric distribution. J Stat Theory Appl. 2019; 18(1): 33-45.

Kemp AW. Classes of discrete lifetime distributions. Commun Stat Theory Methods. 2004; 33(12): 3069-3093.

Khongthip P, Patummasut M, Bodhisuwan W. The discrete weighted exponential distribution and its applications. Songklanakarin J Sci Technol. 2018; 40(5):1105-1114.

Kim HM and Jang YH. New closed-form estimators for weighted Lindley distribution. J Korean Stat Soc. 2021; 50(2): 1-27.

Krishna H, Pundir PS. Discrete Burr and discrete Pareto distributions. Stat Methodol. 2009; 6(2): 177-188.

Lai MT. Optimum number of minimal repairs for a system under increasing failure rate shock model with cumulative repair-cost limit. Int J Reliab Saf. 2013; 7:95-107.

Lindley DV. Fiducial distributions and Bayes’ theorem. J R Stat Soc Series B Stat Methodol. 1958; 102-107.

Mazucheli J, Bertoli W, Oliveira RP. Two useful discrete distributions to model overdispersed count data. Rev Colomb Estad. 2020; 43(1): 21-48.

Mazucheli J, Louzada F, Ghitany ME. Comparison of estimation methods for the parameters of the weighted Lindley distribution. Appl Math Comput. 2013; 220: 463-471.

Nakagawa T, Osaki S. The discrete Weibull distribution. IEEE Trans Reliab. 1975; 24(5): 300-301.

Nooghabi MS, Borzadaran GRM, Roknabadi AHR. Discrete modified Weibull distribution. Metron. 2011; 69(2): 207-222.

Noughabi MS, Rezaei Roknabadi AH, Mohtashami Borzadaran GR. Some discrete lifetime distributions with bathtub-shaped hazard rate functions. Qual Eng. 2013; 25(3): 225-236.

Oliveira RP, Mazucheli J, Santos MLA, Barco KVP. A discrete analogue of the continuous power Lindley distribution and its applications. Revista Brasileira de Biometria. 2018; 36(3): 649-667.

Ridout MS, Besbeas P. An empirical model for under dispersed count data. Stat Modelling. 2004; 4(1): 77-89.

Roy D. Reliability measures in the discrete bivariate set-up and related characterization results for a bivariate geometric distribution. J Multivar Anal. 1993; 46(2): 362-373.

Roy D, Gupta RP. Classifications of discrete lives. Microelectron Reliab. 1992; 32(10): 1459-1473.

Song P, Wang S. A further remark on the alternative expectation formula. Commun Stat Theory Methods. 2019; 50(11): 2586-2591.

Tyagi A, Choudhary N, Singh B. Discrete additive Perks-Weibull distribution: properties and applications. Life Cycle Reliab Saf Eng. 2019; 8(3):183-199.

Xie M, Gaudoin O, Bracquemond C. Redefining failure rate function for discrete distributions. Int J Reliab Qual Saf Eng. 2002; 9(3): 275-285.