We introduce a new generalization of the pseudo-Lindley distribution by applying alpha power transformation. The obtained distribution is referred as the pseudo-Lindley alpha power transformed distribution (PL-APT). Some tractable mathematical properties of the PL-APT distribution as reliability, hazard rate, order statistics and entropies are provided. The maximum likelihood method is used to obtain  the parameters’ estimation of the PL-APT distribution. The asymptotic properties of the proposed distribution are discussed. Also, a simulation study is performed to compare the modeling capability and flexibility of PL-APT with Lindley and pseudo-Lindley distributions. The PL-APT provides a good fit as the Lindley and the pseudo-Lindley distributions. The extremal domain of attraction of PL-APT is found and its quantile and extremal quantile functions are studied. Finally, the extremal value index is estimated by the double-indexed Hill’s estimator (Ngom and Lo [19]) and related asymptotic statistical tests are provided and characterized.

alpha power transformation of distribution functions, Lindley’s distribution, pseudo-Lindley distribution, extreme value theory, doubly indexed Hill’s estimator, reliability, hazard rate, maximum likelihood method, quantile function, extreme quantile function, asymptotic laws, Lambert function.

Received: February 14, 2022; Accepted: March 26, 2022; Published: May 30, 2022

How to cite this article: Modou Ngom, Moumouni Diallo, Adja Mbarka Fall and Gane Samb Lo, The pseudo-Lindley alpha power transformed distribution, mathematical characterizations and asymptotic properties, Far East Journal of Theoretical Statistics 65 (2022), 1-33. http://dx.doi.org/10.17654/0972086322004

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:
[1] M. A. Aldahlan, Alpha power transformed log-logistic distribution with application to breaking stress data, Adv. Math. Phys. 2020 (2020), Article ID 2193787, 9 pages. https://doi.org/10.1155/2020/2193787.
[2] E. Deme, G. S. Lo and A. Diop, On the generalized Hill process for small parameters and applications, J. Stat. Theory Appl. 11(4) (2012), 397-418. http://dx.doi.org/10.2991/jsta.2013.12.1.3.
[3] J. T. Eghwerido, The alpha power Teissier distribution and its applications, Afrika Statistika 16(2) (2021), 2731-2745.
DOI: http://dx.doi.org/10.16929/as/2021.2731.181.
[4] M. E. Ghitany, D. K. Al-Mutairi and S. M. Aboukhamseen, Estimation of the reliability of a stress-strength system from power Lindley distributions, Comm. Statist. Simulation Comput. 44 (2015), 118-136.
[5] E. Gomez-Déniz, M. A. Sordo and E. Calderín-Ojeda, The log-Lindley distribution as an alternative to the beta regression model with applications in insurance, Insurance Math. Econom. 54 (2014), 49-57.
[6] E. H. Hafez, F. H. Riad, Sh. A. M. Mubarak and M. S. Mohamed, Study on Lindley distribution accelerated life tests: application and numerical simulation, Symmetry 12 (2020), 2080. https://doi.org/10.3390/sym12122080.
[7] B. Hill, A simple general approach to the inference about the tail index of a distribution, Ann. Statist. 3(5) (1975), 1163-1174.
[8] M. Ijaz, W. K. Mashwani, A. Göktaş and Y. A. Unvan, A novel alpha power transformed exponential distribution with real-life applications, J. Appl. Stat. 48(11) (2021), 1-16. https://doi.org/10.1080/02664763.2020.1870673.
[9] M. R. Irshad, C. Chesneau, V. D’cruz and R. Maya, Discrete pseudo Lindley distribution: properties, estimation and application on INAR(1) process, Math. Comput. Appl. 26 (2021), 76. https://doi.org/10.3390/mca26040076.
[10] H. Krishna and K. Kumar, Reliability estimation in Lindley distribution with progressively type II right censored sample, Math. Comput. Simulation 82(2) (2011), 281-294.
[11] G. S. Lo, M. Ngom and T. A. Kpanzou, Weak convergence (IA), Sequences of random vectors, 2016. ArXiv: 1810.01625.
DOI: 10.16929/sbs/2016.0001.
[12] G. S. Lo, M. Ngom and M. Diallo, Extremes, extremal index estimation, records, moment problem for the pseudo-Lindley distribution and applications, European Journal of Pure and Applied Mathematics (EJPAM) 13(4) (2020), 739-757.
https://www.ejpam.com; DOI: https://doi.org/10.29020/nybg.ejpam.v13i4.3834.
[13] G. S. Lo, T. A. Kpanzou and C. M. Haidara, Statistical tests for the pseudo-Lindley distribution and applications, Afrika Statistika 14(4) (2019), 2127-2139. DOI: http://dx.doi.org/10.16929/as/2019.2127.151.
[14] G. S. Lo, Mathematical Foundation of Probability Theory, SPAS Books Series, Saint-Louis, Senegal - Calgary, Canada, 2018.
DOI: http://dx.doi.org/10.16929/sbs/2016.0008. Arxiv: arxiv.org/pdf/1808.01713.
[15] D. V. Lindley, Fiducial distributions and Bayes’ theorem, J. Roy. Statist. Soc. Ser. B 20 (1958), 102-107.
[16] D. V. Lindley, Introduction to Probability and Statistics from a Bayesian Viewpoint, Part II: Inference, Cambridge University Press, New York, 1965.
[17] J. Mazucheli and J. A. Achcar, The Lindley distribution applied to competing risks lifetime data, Computer Methods Programs in Biomedicine 104(2) (2011), 188-192.
[18] A. Mahdavi and D. Kundu, A new method for generating distribution with an application to exponential distribution, Comm. Statist. Theory Methods 46(13) (2017), 6543-6557.
[19] M. Ngom and G. S. Lo, A double-indexed functional Hill process and applications, Journal of Mathematical Research 8(4) (2016), 144-165.
DOI: 105539/jmr/v8n4p144.
[20] P. U. Unyime and H. E. Ette, Alpha power transformed quasi Lindley distribution, International Journal of Advanced Statistics and Probability 9(1) (2021), 6-17.
[21] R. Shanker, F. Hagos and S. Sujatha, On modeling of lifetimes data using exponential and Lindley distributions, Biometrics and Biostatistics International Journal 2(5) (2015), 140-147.
[22] R. Shanker and A. Mishra, A quasi Lindley distribution, African Journal of Mathematics and Computer Science Research 6(4) (2013), 64-71.
https://doi.org/10.5897/AJMCSR12.067.
[23] H. Zeghdoudi and S. Nedjar, A pseudo Lindley-distribution and its application, Afrika Statistika 11(1) (2016), 923-932.
DOI: http://dx.doi.org/10.16929/as/2016.923.83.
[24] R. A. ZeinEldin, M. A. U. Haq, S. Hashmi and M. Elsehety, Alpha power transformed inverse Lomax distribution with different methods of estimation and applications, Complexity 2020 (2020), Article ID 1860813, 15 pages. https://doi.org/10.1155/2020/1860813.