Bayesian inference is one of the most important issues under the model selection procedures in statistics. This paper performed a Bayesian analysis of posteriors using a well-known Weibull model with high fitting performance. We proved that there exist necessary and sufficient conditions for the priors to yield proper posteriors and finite posterior moments based on record data. Moreover, we found a significant implication through different well-known objective priors. Finally, for illustration purposes, a Monte Carlo simulation procedure is done in the paper. The results indicate that the developed inference may have a significant contribution within Bayesian analysis through applications across different areas.

References

• Arnold, B. C., Balakrishnan, N., & Nagaraja, H. N. (2011). Records. John Wiley & Sons.
• Balakrishnan, N., Balasubramanian, K., & Panchapakesan, S. (1996). δ-exceedance records. Journal of Applied Statistical Science, 4(2/3), 123–132.
• Benestad, R. E. (2003). How often can we expect a record event? Climate Research, 25(1), 3–13. https://doi.org/10.3354/cr025003
• Berger, J. O., Bernardo, J. M., & Sun, D. (2015). Overall objective priors. Bayesian Analysis, 10(1), 189–221.
• Bernardo, J. M. (1979). Reference posterior distributions for Bayesian inference. Journal of the Royal Statistical Society: Series B (Methodological), 41(2), 113–128.
• Bernardo, J. M. (2005). Reference analysis. Handbook of Statistics, 25, 17–90. https://doi.org/10.1016/S0169-7161(05)25002-2
• Chandler, K. N. (1952). The distribution and frequency of record values. Journal of the Royal Statistical Society: Series B (Methodological), 14(2), 220–228.
• Doostparast, M. (2011). Goodness-of-fit tests for Weibull populations on the basis of records. arXiv preprint arXiv:1110.5509.
• Dziubdziela, W., & Kopociński, B. (1976). Limiting properties of the kth record values. Applicationes Mathematicae, 15(2), 187–190. https://doi.org/10.4064/am-15-2-187-190
• Empacher, C., Kamps, U., & Volovskiy, G. (2023). Statistical prediction of future sports records based on record values. Stats, 6(1), 131–147. https://doi.org/10.3390/stats6010008
• Gembris, D., Taylor, J. G., & Suter, D. (2007). Evolution of athletic records: Statistical effects versus real improvements. Journal of Applied Statistics, 34(5), 529–545. https://doi.org/10.1080/02664760701234850
• Glick, N. (1978). Breaking records and breaking boards. The American Mathematical Monthly, 85(1), 2–26. https://doi.org/10.1080/00029890.1978.11994501
• Gugushvili, S., & Spreij, P. (2014). Nonparametric Bayesian drift estimation for multidimensional stochastic differential equations. Lithuanian Mathematical Journal, 54(2), 127–141. https://doi.org/10.1007/s10986-014-9232-1
• Hassan, A. S., Nagy, H. F., Muhammed, H. Z., & Saad, M. S. (2020). Estimation of multicomponent stress-strength reliability following Weibull distribution based on upper record values. Journal of Taibah University for Science, 14(1), 244–253. https://doi.org/10.1080/16583655.2020.1721751
• Jafari, A. A., & Bafekri, S. (2021). Inferences on the performance index of Weibull distribution based on k-record values. Journal of Computational and Applied Mathematics, 382, 113060. https://doi.org/10.1016/j.cam.2020.113060
• Juhás, M., & Skřivánková, V. 2014. Characterization of general classes of distributions based on independent property of transformed record values. Applied Mathematics and Computation 226 44–50. https://doi.org/10.1016/j.amc.2013.10.037
• Kang, S. G., Lee, W. D., & Kim, Y. (2017). Noninformative priors for the ratio of the shape parameters of two Weibull distributions. Computational Statistics, 32(1), 35–50. https://doi.org/10.1007/s00180-015-0631-5
• Kass, R. E., & Wasserman, L. (1996). The selection of prior distributions by formal rules. Journal of the American Statistical Association, 91(435), 1343–1370. https://doi.org/10.1080/01621459.1996.10477003
• Kauffman, S., & Levin, S. (1987). Towards a general theory of adaptive walks on rugged landscapes. Journal of Theoretical Biology, 128(1), 11–45. https://doi.org/10.1016/S0022-5193(87)80029-2
• Kim, Y., & Seo, J. I. (2020). objective Bayesian prediction of future record statistics based on the exponentiated Gumbel distribution: Comparison with time-series prediction. Symmetry, 12(9), 1443. https://doi.org/10.3390/sym12091443
• Kizilaslan, F., & Nadar, M. (2017). Statistical inference of P(X<Y) for the Burr Type XII distribution based on records. Hacettepe Journal of Mathematics and Statistics, 46(4), 713–742.
• Lee, W. D., Kang, S. G., & Kim, Y. (2015). Noninformative priors for the common shape parameters of Weibull distributions. Journal of the Korean Statistical Society, 44(4), 668–679. https://doi.org/10.1016/j.jkss.2015.07.003
• Lee, W. D., Kang, S. G., & Kim, Y. (2017). Objective Bayesian inference for the ratio of the scale parameters of two Weibull distributions. Communications in Statistics-Theory and Methods, 46(10), 4943–4956. https://doi.org/10.1080/03610926.2015.1091477
• Liang, F., Paulo, R., Molina, G., Clyde, M. A., & Berger, J. O. (2008). Mixtures of g priors for Bayesian variable selection. Journal of the American Statistical Association, 103(481), 410–423. https://doi.org/10.1198/016214507000001337
• Madadi, M., & Tata, M. (2014). Shannon information in k-records. Communications in Statistics – Theory and Methods, 43(15), 3286–3301. https://doi.org/10.1080/03610926.2012.697965
• Martinez, E. Z., de Freitas, B. C. L., Achcar, J. A., Aragon, D. C., & de Oliveira Peres, M. V. (2022). Exponentiated Weibull models applied to medical data in presence of right-censoring, cure fraction and covariates. Statistics, Optimization & Information Computing, 10(2), 548–571. https://doi.org/10.19139/soic.v10i2
• Nasiri, P., & Hosseini, S. (2012). Statistical inferences for Lomax distribution based on record values (Bayesian and classical). Journal of Modern Applied Statistical Methods, 11(1), 179–189. https://doi.org/10.22237/jmasm/1335845640
• Neal, P., & Roberts, G. (2008). Optimal scaling for random walk Metropolis on spherically constrained target densities. Methodology and Computing in Applied Probability, 10(2), 277–297. https://doi.org/10.1007/s11009-007-9046-2
• Nevzorov, V. B. (2000). Records: Mathematical theory. AMS.
• Northrop, P. J., & Attalides, N. (2016). Posterior propriety in Bayesian extreme value analyses using reference priors. Statistica Sinica, 26(2), 721–743.
• R Core Team (2024). R: A language and environment for statistical computing.
• Ramos, E., Egbon, O. A., Ramos, P. L., Rodrigues, F. A., & Louzada, F. (2020). Objective Bayesian analysis for the differential entropy of the Gamma distribution. arXiv preprint arXiv:2012.14081.
• Ramos, E., Ramos, P. L., & Louzada, F. (2020). Posterior properties of the Weibull distribution for censored data. Statistics & Probability Letters, 166, 108873. https://doi.org/10.1016/j.spl.2020.108873
• Ramos, P. L., Achcar, J. A., Moala, F. A., Ramos, E., & Louzada, F. (2017). Bayesian analysis of the generalized gamma distribution using non-informative priors. Statistics, 51(4), 824–843.
• Ramos, P. L., Almeida, M. H., Louzada, F., Flores, E., & Moala, F. A. (2022). Objective Bayesian inference for the Capability index of the Weibull distribution and its generalization. Computers & Industrial Engineering, 167, 108012. https://doi.org/10.1016/j.cie.2022.108012
• Ramos, P. L., Dey, D. K., Louzada, F., & Ramos, E. (2021). On posterior properties of the two parameter gamma family of distributions. Anais da Academia Brasileira de Ciencias, 93(suppl 3), e20190826. https://doi.org/10.1590/0001-3765202120190826
• Ramos, P. L., Rodrigues, F. A., Ramos, E., Dey, D. K., & Louzada, F. (2023). Power laws distributions in objective priors. Statistica Sinica, 33(3), 1–53.
• Robert, C. P., & Casella, G. (2010). Introducing Monte Carlo methods with R (Vol. 18). Springer.
• Shakhatreh, M. K., Dey, S., & Alodat, M. T. (2021). Objective Bayesian analysis for the differential entropy of the Weibull distribution. Applied Mathematical Modelling, 89, 314–332. https://doi.org/10.1016/j.apm.2020.07.016
• Shakhatreh, M. K., Lemonte, A. J., & Cordeiro, G. M. (2020). On the generalized extended exponential-Weibull distribution: Properties and different methods of estimation. International Journal of Computer Mathematics, 97(5), 1029–1057. https://doi.org/10.1080/00207160.2019.1605062
• Sun, D. (1997). A note on noninformative priors for Weibull distributions. Journal of Statistical Planning and Inference, 61(2), 319–338. https://doi.org/10.1016/S0378-3758(96)00155-3
• Syam, A. H., & Barakat, H. M. (2022). Information measures for record values and their concomitants under Haung-Kotz FGM bivariate distribution. Bulletin of Faculty of Science, Zagazig University, 2022(3), 122–130. https://doi.org/10.21608/bfszu.2022.147965.1154
• Teimouri, M., & Nadarajah, S. (2013). Bias corrected MLEs for the Weibull distribution based on records. Statistical Methodology, 13, 12–24. https://doi.org/10.1016/j.stamet.2013.01.001
• Thornton, C., Hutter, F., Hoos, H. H., & Leyton-Brown, K. (2013). Auto-WEKA: Combined selection and hyperparameter optimization of classification algorithms. In Proceedings of the 19th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (pp. 847–855).
• Vidović, Z. (2021). Random chord in a circle and Bertrand's paradox: New generation method, extreme behaviour and length moments. Bulletin of the Korean Mathematical Society, 58(2), 433–444.
• Volovskiy, G., & Kamps, U. (2023). Likelihood-based prediction of future Weibull record values. REVSTAT-Statistical Journal, 21(3), 425–445.
• Wang, B. X., & Ye, Z. S. (2015). Inference on the Weibull distribution based on record values. Computational Statistics & Data Analysis, 83, 26–36. https://doi.org/10.1016/j.csda.2014.09.005
• Weibull, W. (1951). A statistical distribution function of wide applicability. Journal of Applied Mechanics, 18(3), 293–297. https://doi.org/10.1115/1.4010337
• Wergen, G. (2013). Records in stochastic processes: Theory and applications. Journal of Physics A: Mathematical and Theoretical, 46(22), 223001. https://doi.org/10.1088/1751-8113/46/22/223001
• Xia, Z. P., Yu, J. Y., Cheng, L. D., Liu, L. F., & Wang, W. M. (2009). Study on the breaking strength of jute fibres using modified Weibull distribution. Composites Part A: Applied Science and Manufacturing, 40(1), 54–59. https://doi.org/10.1016/j.compositesa.2008.10.001
• Zellner, A. (1977). Maximal data information prior distributions. New developments in the applications of Bayesian methods (pp. 211–232). North-Holland.

   Google Scholar

To cite this article: Z. Vidović, J. Nikolić & Z. Perić (2024) Properties of k-record posteriors for the Weibull model, Statistical Theory and Related Fields, 8:2, 152-162, DOI: 10.1080/24754269.2024.2322312

To link to this article: https://doi.org/10.1080/24754269.2024.2322312