Classical and Bayesian Inference for the Length Biased Weighted Lomax Distribution under Progressive Censoring Scheme

Abstract

In this study, the length biassed weighted Lomax (LBWLo) distribution's reliability and hazard functions, as well as the population characteristics, are evaluated using progressively Type II censored samples. The proposed estimators are obtained by combining the maximum likelihood and Bayesian approaches. The posterior distribution of the LBWLo distribution is derived from the Gamma and Jeffery's priors, which, respectively, act as informative and non-informative priors. The Metropolis-Hasting (MH) algorithm is also utilized to get the Bayesian estimates. Based on the Fisher information matrix, we derive asymptotic confidence intervals. We create the intervals with the highest posterior density using the sample the MH technique generated. Numerical simulation research is done to evaluate the effectiveness of the approaches. Through Monte Carlo simulation, we compare the proposed estimates in terms of mean squared error. It is possible to get coverage probability and average interval lengths of 95%. The study's findings supported the idea that, in the majority of the cases, Bayes estimates with an informative prior are more appropriate than other estimates. Additionally, one set of actual data supported the findings of the study.

Keywords

• Length biased weighted Lomax
• Maximum likelihood estimation
• squared error loss function
• informative prior
• Markov Chain Monte Carlo

References

1. [1] Lomax, K. S., “Business failures: Another example of the analysis of failure data”, Journal of the American Statistical Association, 49(268): 847–852, (1954).
2. [2] Harris, C. M., “The Pareto distribution as a queue service discipline”, Operations Research, 16 (2): 307–313, (1968).
3. [3] Atkinson, A.B., and Harrison, A.J., “Distribution of Personal Wealth in Britain”, Cambridge University Cambridge, (1987).
4. [4] Holland, O., Golaup, A., and Aghvami, A. H., “Traffic characteristics of aggregated module downloads for mobile terminal reconfiguration”, IEE Proceedings- Communications, 153(5): 683–690, (2006). http: //dx.doi.org/10.1049/ip-com:20045155
5. [5] Hassan, A. S., and Al-Ghamdi, A. S., “Optimum step stress accelerated life testing for Lomax distribution”, Journal of Applied Sciences Research, 5: 21532164, (2009).
6. [6] Hassan, A. S., Assar, S. M., and Shelbaia, A., “Optimum step stress accelerated life test plan for Lomax distribution with an adaptive type-II progressive hybrid censoring”, British Journal of Mathematics and Computer Science, 13(2): 119, (2016).
7. [7] Hassan, A.S., and Mohamed, R.E, “ Parameter estimation of inverse exponentiated Lomax with right censored data”, Gazi University Journal of Science, 32(4): 13701386,(2019).
8. [8] Muhammad, IJAZ, “ Bayesian estimation of the shape parameter of Lomax distribution under uniform and Jeffery prior with engineering applications”, Gazi University Journal of Science, 34(2): 562577, (2021).

9. [9] Hassan, A.S., and Ismail, D, “Estimation of parameters of Topp-Leone inverse Lomax distribution in presence of right censored samples”, Gazi University Journal of Science, 34(4): 11931208, (2021).
10. [10] Ahmad, A., Ahmad, S.P., and Ahmed, A., Length-biased weighted Lomax distribution: statistical properties and application”, Pakistan Journal of Statistics and Operation Research, 12: 245-255, (2016).
11. [11] Karimi, H., and Nasiri, P., “Estimation parameter of R = P(Y < X) for length-biased weighted Lomax distributions in the presence of outliers,” Mathematical and Computational Applications, 23(9): 1–9, (2018).
12. [12] Bantan, R., Hassan, A.S., Almetwally, E., Elgarhy, M. Jamal, F., Chesneau, C., and Elsehetry, M., “Bayesian analysis in partially accelerated life tests for weighted Lomax distribution”, Computers, Materials & Continua, 68 (3): 28592875, (2021).
13. [13] Hofmann, G., Cramer, E., Balakrishnan, N., and Kunert, G., “An asymptotic approach to progressive censoring”, Journal of Statistical Planning and Inference, 130: 207–227, (2005).
14. [14] Krishna, H., and Kumar, K., “Reliability estimation in Lindley distribution with progressively type II right censored sample”, Mathematics and Computers in Simulation, 82: 281–294, (2011).
15. [15] Kohansal, A.,“ Statistical analysis of two-parameter bathtub-shaped lifetime distribution under progressive censoring with binomial removals”, Gazi University Journal of Science, 29(4): 783-792, (2016).
16. [16] Cetinkaya, C., “Estimation in step-stress partially accelerated life tests for the power Lindley distribution under progressive censoring”, Gazi University Journal of Science, 34(2): 579590, (2021).
17. [17] Shrahili, M., El-Saeed, A.R., Hassan, A.S., Elbatal, I., and Elgarhy, M., “Estimation of entropy for log-Logistic distribution under progressive type II censoring”, Journal of Nanomaterials, (2022). doi.org/10.1155/2022/2739606
18. [18] Hassan, A. S., Mousa, R. M., and Abu-Moussa, M. H., “Analysis of progressive type-II competing risks data with applications”, Lobachevskii Journal of Mathematics, 43(9): 2479–2492, (2022).
19. [19] Akdogan, Y., Kus, C., and Wu,S-J.,“ Planning life tests for Burr XII distributed products under progressive group-censoring with cost considerations”, Gazi University Journal of Science, 25(2): 425-434 (2012).
20. [20] Balakrishnan, N., and Aggarwala, R., “Progressive Censoring Theory, Methods and Applications”, Birkhauser Boston, MA, (2000).
21. [21] Cohen, A. C., “Maximum likelihood estimation in the Weibull distribution based on complete and censored samples”, Technometrics, 7: 579588, (1965).
22. [22] Lawless, J.F., “Statistical models and methods for lifetime data”, Wiley, New York, (1982).
23. [23] Greene, W.H., “Econometric analysis”, 4th edn. Prentice-Hall, New York, (2000).
24. [24] Balakrishnan, N., and Sandhu, R.A., “A simple simulation algorithm for generating progressively type II censored samples”, American Statistical Association, 49 (2): 229–230, (1995).
25. [25] Dey, S., and Pradhan, B., “Generalized inverted exponential distribution under hybrid censoring”, Statistical Methodology, 18: 101114, (2014).
26. [26] Dey, S., Singh, S., Tripathi, Y.M., and Asgharzadeh, A., “Estimation and prediction for a progressively censored generalized inverted exponential distribution”, Statistical Methodology, 32: 185202, (2016).
27. [27] Hassan, A. S., and Assar, S. M., “The exponentiated Weibull-power function distribution. Journal of Data Sciences”, 15(4): 589-614, (2017).
28. [28] Gadde, S.R., and Al-Omari, A.I., “Attribute control charts based on TLT for length-biased weighted Lomax distribution”, Journal of Mathematics, (2022).