Review Articles

Properties of k-record posteriors for the Weibull model

Z. Vidović ,

Faculty of Education, University of Belgrade, Belgrade, Serbia

zoran.vidovic@uf.bg.ac.rs

J. Nikolić ,

Faculty of Electronic Engineering, University of Niš, Niš, Serbia

Z. Perić

Faculty of Electronic Engineering, University of Niš, Niš, Serbia

Pages | Received 11 Sep. 2023, Accepted 15 Feb. 2024, Published online: 04 Mar. 2024,
  • Abstract
  • Full Article
  • References
  • Citations

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 Science4(2/3), 123–132.
  • Benestad, R. E. (2003). How often can we expect a record event? Climate Research25(1), 3–13. https://doi.org/10.3354/cr025003
  • Berger, J. O., Bernardo, J. M., & Sun, D. (2015). Overall objective priors. Bayesian Analysis10(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 Statistics25, 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 Mathematicae15(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. Stats6(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 Statistics34(5), 529–545. https://doi.org/10.1080/02664760701234850
  • Glick, N. (1978). Breaking records and breaking boards. The American Mathematical Monthly85(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 Journal54(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 Science14(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 Mathematics382, 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 Statistics32(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 Association91(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 Biology128(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. Symmetry12(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 Statistics46(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 Society44(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 Methods46(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 Association103(481), 410–423. https://doi.org/10.1198/016214507000001337
  • Madadi, M., & Tata, M. (2014). Shannon information in k-records. Communications in Statistics – Theory and Methods43(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 Computing10(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 Methods11(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 Probability10(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 Sinica26(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 Letters166, 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. Statistics51(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 Engineering167, 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 Ciencias93(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 Sinica33(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 Modelling89, 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 Mathematics97(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 Inference61(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 University2022(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 Methodology13, 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 Society58(2), 433–444.
  • Volovskiy, G., & Kamps, U. (2023). Likelihood-based prediction of future Weibull record values. REVSTAT-Statistical Journal21(3), 425–445.
  • Wang, B. X., & Ye, Z. S. (2015). Inference on the Weibull distribution based on record values. Computational Statistics & Data Analysis83, 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 Mechanics18(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 Theoretical46(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 Manufacturing40(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.

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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