Unit Power Lindley Distribution: Properties and Estimation

Abstract

This paper introduces the unit power Lindley distribution and presents its fundamental statistical properties, such as density and cumulative distribution functions, hazard rate functions, and, their characteristics, moments, and related measures. The parameters of this newly proposed distribution are estimated by using six different methods: maximum likelihood, least squares, weighted least squares, Cramér von Mises, Anderson Darling, and right-tail Anderson Darling. The performances of the considered estimation methods are compared through an extensive Monte Carlo simulation study. Additionally, two real datasets are modeled to demonstrate that the unit power Lindley distribution provides a significantly better fit than compared to commonly used unit distributions.

Keywords

• Unit power Lindley
• Estimation
• Unit distributions

References

1. [1] Cook, D.O., Kieschnick, R., and McCullough, B.D., “Regression analysis of proportions in finance with self selection”, Journal of Empirical Finance, 15(5): 860-867, (2008).
2. [2] Cribari-Neto, F., and Souza, T.C. “Religious belief and intelligence: Worldwide evidence”, Intelligence, 41(5): 482-489, (2013).
3. [3] Gupta, A.K., and Nadarajah, S., Handbook of Beta Distribution and Its Applications, Marcel Dekker, New York, (2004).
4. [4] Hunger, M., Baumert, J., and Holle, R., “Analysis of SF-6D index data: is beta regression appropriate?”, Value in Health, 14(5): 759-767, (2011).
5. [5] Kieschnick, R., and McCullough, B.D., “Regression analysis of variates observed on (0, 1): percentages, proportions and fractions”, Statistical Modelling, 3(3): 193-213, (2003).
6. [6] Papke, L.E., and Wooldridge, J.M., “Econometric methods for fractional response variables with an application to 401 (k) plan participation rates”, Journal of Applied Econometrics, 11(6): 619-632, (1996).
7. [7] Souza, T.C., and Cribari-Neto, F., “Intelligence, religiosity and homosexuality non-acceptance: Empirical evidence”, Intelligence, 52: 63-70, (2015).
8. [8] Mazucheli, J., Menezes, A.F.B., and Chakraborty, S., “On the one parameter unit-Lindley distribution and its associated regression model for proportion data”, Journal of Applied Statistics, 46(4): 700-714, (2019).

9. [9] Topp, C.W., and Leone, F.C., “A family of J-shaped frequency functions”, Journal of the American Statistical Association, 50(269): 209-219, (1955).
10. [10] Kumaraswamy, P., “A generalized probability density function for double-bounded random processes”, Journal of Hydrology, 46(1-2): 79-88, (1980).
11. [11] Nadarajah, S., and Kotz, S., “Moments of some J-shaped distributions”, Journal of Applied Statistics, 30(3): 311-317, (2003).
12. [12] Cordeiro, G.M., and Castro, M.D., “A new family of generalized distributions”, Journal of Statistical Computation and Simulation, 81(7): 883-898, (2011).
13. [13] Consul, P.C., and Jain, G.C., “On the log-gamma distribution and its properties”, Statistische Hefte, 12(2): 100-106, (1971).
14. [14] Mazucheli, J., Menezes, A.F.B. and Ghitany, M.E., “The unit-Weibull distribution and associated inference”, Journal of Applied Probability and Statistics, 13(2): 1-22, (2018).
15. [15] Mazucheli, J., Menezes, A.F.B. and Dey, S., “The unit-Birnbaum-Saunders distribution with applications”, Chilean Journal of Statistics, 9(1): 47-57, (2018).
16. [16] Ghitany, M.E., Mazucheli, J., Menezes, A.F.B. and Alqallaf, F., “The unit-inverse Gaussian distribution: A new alternative to two-parameter distributions on the unit interval”, Communications in Statistics-Theory and Methods, 48(14): 3423-3438, (2019).
17. [17] Mazucheli, J., Bapat, S.R. and Menezes, A.F.B., “A new one-parameter unit-Lindley distribution”, Chilean Journal of Statistics, 11(1): 53-67, (2020).
18. [18] Korkmaz, M.Ç. and Chesneau, C., “On the unit Burr-XII distribution with the quantile regression modeling and applications”, Computational and Applied Mathematics, 40: 1:29, (2021).
19. [19] Korkmaz, M.Ç., Altun, E., Chesneau, C. and Yousof, H.M., “On the unit-Chen distribution with associated quantile regression and applications”, Mathematica Slovaca, 72(3): 765-786, (2022).
20. [20] Haq, M.A.U., Hashmi, S., Aidi, K., Ramos, P.L. and Louzada, F., “Unit modified Burr-III distribution: Estimation, characterizations and validation test”, Annals of Data Science, 10(2): 415-440, (2023).
21. [21] Korkmaz, M.Ç. and Korkmaz, Z.S., “The unit log–log distribution: A new unit distribution with alternative quantile regression modeling and educational measurements applications”, Journal of Applied Statistics, 50(4): 889-908, (2023).
22. [22] Cakmakyapan, S. and Ozel, G., “The Lindley family of distributions: properties and applications”, Hacettepe Journal of Mathematics and Statistics, 46(6): 1113-1137, (2016).
23. [23] Ozel, G., Alizadeh, M., Cakmakyapan, S., Hamedani, G. G., Ortega, E. M. and Cancho, V.G., “The odd log-logistic Lindley Poisson model for lifetime data”, Communications in Statistics-Simulation and Computation, 46(8): 6513-6537, (2017).
24. [24] Cordeiro, G.M., Afify, A.Z., Yousof, H.M., Cakmakyapan, S. and Ozel, G., “The Lindley Weibull distribution: properties and applications”, Anais da Academia Brasileira de Ciências, 90: 2579-2598, (2018).
25. [25] Cakmakyapan, S. and Ozel, G., “Generalized Lindley family with application on wind speed data”, Pakistan Journal of Statistics and Operation Research, 17(2): 387-397, (2021).
26. [26] Ghitany, M.E., Al-Mutairi, D.K., Balakrishnan, N. and Al-Enezi, L.J., “Power Lindley distribution and associated inference”, Computational Statistics & Data Analysis, 64: 20-33, (2013).
27. [27] Çakmak, B. and Doğru, F.Z., “Optimal B-robust estimators for the parameters of the power Lindley distribution”, Journal of Applied Statistics, 48(13-15): 2369-2388, (2021).
28. [28] Arslan, T., Acitas, S. and Senoglu, B., “Generalized Lindley and power Lindley distributions for modeling the wind speed data”, Energy Conversion and Management, 152: 300-311, (2017).
29. [29] Pak, A., Gupta, A.K. and Khoolenjani, N.B., “On reliability in a multicomponent stress-strength model with power Lindley distribution”, Revista Colombiana de Estadistica, 41(2): 251-267, (2018).
30. [30] Kumar, S., Yadav, A.S., Dey, S. and Saha, M., “Parametric inference of generalized process capability index Cpyk for the power Lindley distribution”, Quality Technology Quantitative Management, 19(2): 153-186, (2022).
31. [31] Sharma, V.K., Singh, S.K. and Singh, U., “Classical and Bayesian methods of estimation for power Lindley distribution with application to waiting time data”, Communications for Statistical Applications and Methods, 24(3): 193-209, (2017).
32. [32] Valiollahi, R., Raqab, M.Z., Asgharzadeh, A. and Alqallaf, F.A., “Estimation and prediction for power Lindley distribution under progressively type II right censored samples”, Mathematics and Computers in Simulation, 149: 32-47, (2018).
33. [33] Kumar, D. and Goyal, A., “Order statistics from the power Lindley distribution and associated inference with application”, Annals of Data Science, 6(1): 153-177, (2019).
34. [34] Lindley, D.V., “Fiducial distributions and Bayes' theorem”, Journal of the Royal Statistical Society. Series B (Methodological), 102-107, (1958).
35. [35] Ghitany, M.E., Atieh, B. and Nadarajah, S., “Lindley distribution and its application”, Mathematics and Computers in Simulation, 78(4): 493-506, (2008).
36. [36] Chapeau-Blondeau, F. and Monir, A., “Numerical evaluation of the Lambert W function and application to generation of generalized Gaussian noise with exponent ½”, IEEE Transactions on Signal Processing, 50(9): 2160-2165, (2002).
37. [37] Swain, J.J., Venkatraman, S. and Wilson, J.R., “Least-squares estimation of distribution functions in Johnson's translation system”, Journal of Statistical Computation and Simulation, 29(4): 271-297, (1988).
38. [38] Wolfowitz, J., “Estimation by the minimum distance method”, Annals of the Institute of Statistical Mathematics, 5(1): 9-23, (1953).
39. [39] Wolfowitz, J., “The minimum distance method”, The Annals of Mathematical Statistics, 75-88, (1957).
40. [40] Akgül, F.G., “Comparison of the estimation methods for the parameters of exponentiated reduced Kies distribution”, Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 22(3): 1209-1216, (2018).
41. [41] Lagarias, J.C., J.A. Reeds, M.H. Wright, and P.E. Wright., “Convergence properties of the Nelder-Mead simplex method in low dimensions”, SIAM Journal on Optimization, 9(1): 112–147, (1998).
42. [42] Dasgupta, R., “On the distribution of burr with applications”, Sankhya B, 73: 1-19, (2011).
43. [43] Akaike, H., “Information theory and an extension of the maximum likelihood principle, In selected papers of hirotugu akaike”, NY: Springer New York, (1998).
44. [44] Schwarz, G., “Estimating the dimension of a model”, The Annals of Statistics, 461-464, (1978).
45. [45] Bai, Z.D., Krishnaiah, P.R. and Zhao, L.C., “On rates of convergence of efficient detection criteria in signal processing with white noise”, IEEE Transactions on Information Theory, 35(2): 380-388, (1989).