Review Articles

Bayesian analysis for the Lomax model using noninformative priors

Asish Banik ,

Department of Statistics & Probability, Michigan State University, East Lansing, MI, USA

banikasi@msu.edu

Taps Maiti ,

Department of Statistics & Probability, Michigan State University, East Lansing, MI, USA

Andrew Bender

Department of Epidemiology & Biostatistics, Department of Neurology & Ophthalmology, Michigan State University, East Lansing, MI, USA

Pages | Received 09 Nov. 2021, Accepted 02 Oct. 2022, Published online: 14 Oct. 2022,
  • Abstract
  • Full Article
  • References
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The Lomax distribution is an important member in the distribution family. In this paper, we systematically develop an objective Bayesian analysis of data from a Lomax distribution. Noninformative priors, including probability matching priors, the maximal data information (MDI) prior, Jeffreys prior and reference priors, are derived. The propriety of the posterior under each prior is subsequently validated. It is revealed that the MDI prior and one of the reference priors yield improper posteriors, and the other reference prior is a second-order probability matching prior. A simulation study is conducted to assess the frequentist performance of the proposed Bayesian approach. Finally, this approach along with the bootstrap method is applied to a real data set.

References

  • Atkinson, A. B., & Harrison, A. J. (1978). Distribution of personal wealth in Britain. Cambridge University Press. 
  • Bain, L. J., & Engelhardt, M. (1992). Introduction to probability and mathematical statistics. PWSKENT Publishing Company. 
  • Berger, J. O., & Bernardo, J. M. (1992). Ordered group reference priors with application to the multinomial problem. Biometrika79(1), 25–37. https://doi.org/10.1093/biomet/79.1.25 
  • Berger, J. O., Bernardo, J. M., & Sun, D. C. (2009). The formal definition of reference priors. The Annals of Statistics37(2), 905–938. https://doi.org/10.1214/07-AOS587
  • Bernardo, J. M. (1979). Reference posterior distributions for Bayesian inference (with discussion). Journal of the Royal Statistical Society: Series B (Methodological)41(2), 113–147. https://doi.org/10.1111/j.2517-6161.1979.tb01066.x 
  • Chakraborty, T. (2019). An analysis of the maximum likelihood estimates for the Lomax distribution. ArXiv: 1911.12612v2, 1–14. 
  • Consonni, G., Fouskakis, D., Liseo, B., & Ntzoufras, I. (2018). Prior distributions for objective Bayesian analysis. Bayesian Analysis13(2), 627–679. https://doi.org/10.1214/18-BA1103
  • Datta, G. S., & Mukerjee, R. (2004). Probability matching priors: Higher order asymptotics. Lecture notes in statistics. Sringer. 
  • Deville, Y. (2016). Renext: Renewal method for extreme values extrapolation (p. 25). 
  • Ferreira, P. H., Gonzales, J. F. B., Tomazella, V. L. D., Ehlers, R. S., Louzada, F., & Silva, E. B. (2016). Objective Bayesian analysis for the Lomax distributionms. ArXiv: 1602.08450v1, 1–19. 
  • Ferreira, P. H., Ramos, E., Ramos, P. L., Gonzales, J. F. B., Tomazella, V. L. D., R. S. Ehlers, Silva, E. B., & Louzad, F. (2020). Objective Bayesian analysis for the Lomax distribution. Statistics & Probability Letters159, Article 108677. https://doi.org/10.1016/j.spl.2019.108677
  • Holland, O., Golaup, A., & Aghvami, A. H. (2006). Traffic characteristics of aggregated module downloads for mobile terminal reconfiguration. IEEE Proceedings: Communications153(5), 683–690. https://doi.org/10.1049/ip-com:20045155 
  • Jeffreys, H. (1961). Theory of probability (3rd ed.). Oxford University Press. 
  • Kang, S. G., Lee, W. D., & Kim, Y. (2021). Posterior propriety of bivariate lomax distribution under objective priors. Communication in Statistics – Theory and Methods50(9), 2201–2209. https://doi.org/10.1080/03610926.2019.1662049 
  • Lemonte, A. J., & Cordeiro, G. M. (2013). An extended Lomax distribution. Statistics47(4), 800–816. https://doi.org/10.1080/02331888.2011.568119 
  • Lindley, D. V. (1956). On a measure of the information provided by an experiment. The Annals of Mathematical Statistics27(4), 986–1005. https://doi.org/10.1214/aoms/1177728069 
  • Lomax, K. (1954). Business failures: Another example of the analysis of failure data. Journal of the American Statistical Association49(268), 847–852. https://doi.org/10.1080/01621459.1954.10501239
  • Marshall, A. W., & Olkin, I. (2007). Life distributions: Structure of nonparametric, semiparametric, and parametric families. Springer. 
  • Nadarajah, S. (2005). Sums, products, and ratios for the bivariate Lomax distribution. Computational Statistics & Data Analysis49(1), 109–129. https://doi.org/10.1016/j.csda.2004.05.003
  • Nayak, T. K. (1987). Multivariate Lomax distribution: Properties and usefulness in reliability theory. Journal of Applied Probability24(1), 170–177. https://doi.org/10.2307/3214068
  • Peers, H. W. (1965). On confidence sets and Bayesian probability points in the case of several parameters. Journal of the Royal Statistical Society: Series B (Methodological)27(1), 9–16. https://doi.org/10.1111/j.2517-6161.1965.tb00581.x 
  • Ramos, P. L., Louzada, F., & Ramos, E. (2018). Posterior properties of the Nakagami-m distribution using non-informative priors and applications in reliability. IEEE Transactions on Reliability67(1), 105–117. https://doi.org/10.1109/TR.24
  • Roberts, G. O., Gelman, A., & Gilks, W. R. (1997). Weak convergence and optimal scaling of random walk metropolis algorithms. Annals of Applied Probability7(1), 110–120 . http://doi.org/10.1214/aoap/1034625254 
  • Roy, D., & Gupta, R. P. (1996). Bivariate extension of Lomax and finite range distributions through characterization approach. Journal of Multivariate Analysis59(1), 22–33. https://doi.org/10.1006/jmva.1996.0052
  • Zellner, A. (1977). Maximal data information prior distributions. In A. Aykac & C. Brumat (Eds.), New developments in the applications of Bayesian methods (pp. 211–232). North-Holland. 

To cite this article: Daojiang He, Dongchu Sun & Qing Zhu (2022): Bayesian analysis for the Lomax model using noninformative priors, Statistical Theory and Related Fields, DOI: 10.1080/24754269.2022.2133466 To link to this article: https://doi.org/10.1080/24754269.2022.2133466