- Author
- Makhdoom Iman (Payame Noor University (PNU), Tehran, Iran), Pak Abbas (Shahrekord University, Shahrekord, Iran)
- Title
- On Bayesian Inference of Reliability Parameter in Burr-type XII Model Based on Imprecise Data: a Survey on Fuzzy Modelling
- Source
- Statistics in Transition, 2024, vol. 25, nr 1, s. 125-143, tab., wykr., bibliogr. 35 poz.
- Keyword
- Estymacja bayesowska, Metoda największej wiarygodności, Łańcuch Markowa, Symulacja Monte Carlo
Bayesian estimation, Maximum likelihood estimation, Markov chain, Monte Carlo simulation - Note
- summ.
- Abstract
- There are always two major sources of uncertainty in measurements related to lifetime surveys: variation among the observations and imprecision of individual observation called fuzziness. The typical statistical analysis is based on variation among the observations and does not consider the imprecision due to individual observation. However, ignoring the imprecision of individual observations may cause losing information and getting misleading results. It is mandatory to analyse such data, to extend the real numbers classically and Bayesian estimation methods to fuzzy numbers. Inference on the Burr-type (BT) XII model, based on precise measurements, is carried out by researchers, yet the problem of estimating parameters, in the presence of fuzzy data, remains unresolved. We are estimating the BT XII distribution parameters and their corresponding reliability when the available data are in the fuzzy numbers. The maximum likelihood estimation (MLE), the Bayesian method and the method of moments are used for estimating parameters. Finally, these estimators are compared via a Monte-Carlo simulation study. (original abstract)
- Accessibility
- The Main Library of the Cracow University of Economics
- Full text
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- Bibliography
- Akbari, M. G., Rezaei, A., (2007). A uniformly minimum variance unbiased point estimator using fuzzy observations. Austrian Journal of Statistics, 36 (4), pp. 307-317.
- Ali Mousa, M. A. M., Jaheen, Z. F., (2002). Statistical inference for the Burr model based on progressively censored data functions. Comput. Math. Applic., 43, pp. 1441- 1449.
- Burr, I. W., (1942). Cumulative frequency functions. Annals of Mathematical Statistics, 13 , pp. 215-222.
- Burr, I. W., (1973). Parameters for a general system of distributions to match a grid of ?3and ?4. Communications in Statistics-Theory and Methods, 2 , pp. 1-21.
- Coppi, R., Gilb, M. A., Kiersc, H. A. L., (2006). The fuzzy approach to statistical analysis. Computational Statistics and Data Analysis, 51 (1), pp. 1-14.
- Denoeux, T., (2011). Maximum likelihood estimation from fuzzy data using the EM algorithm, Fuzzy Sets and Systems, doi:10.1016/j.fss.2011.05.022.
- Gertner, G. Z., Zhu, H., (1996). Bayesian estimation in forest surveys when samples or prior information are fuzzy. Fuzzy Sets and Systems., 77(3), pp. 277-290.
- Gil, M. A., López-Diaz, M., Ralescu, D. A., (2006). Overview on the development of fuzzy random variables. Fuzzy Sets and Systems, 157, pp. 2546-2557.
- Hanagl, D. D., Ahmadi, K. A., (2009). Bayesian estimation of the parameters of bivariate exponential distribution, Communication in Statistics-Simulation and Computation, 38, pp. 1391-1413.
- Hate, M. A., (1949). A certain cumulative probability function. Ann. Math. Statist., 20, pp. 461-63.
- Huang, H., Zuo, M., Sun, Z., (2006). Bayesian reliability analysis for fuzzy lifetime data. Fuzzy Sets and Systems, 157(3), pp. 1674-1686.
- Moore, D., Papadopoulos, A. S., (2000). The Burr type XII distribution as a failure model under various loss functions. Microelectron. Reliab., 40, pp. 2117-2122, doi: 10.1016/S0026-2714(00)00031-7.
- Pak, A., Parham, G. H., Saraj, M., (2013). Inference for the Weibull Distribution Based on Fuzzy Data. Revista Colombiana de Estadistica, 36(2), pp. 339-358.
- Pak, A., Parham, G. H., Saraj, M., (2014). Inferences on the Competing Risk Reliability Problem for Exponential Distribution Based on Fuzzy Data. IEEE Transactions on reliability, 63(1), pp. 2-13.
- Pak, A., (2016). Statistical Inference for the Parameter of Lindley Distribution Based on Fuzzy data. To appear in Brazilian Journal of Probability and Statistics.
- Rodriguez, R. N., (1977). A guide to the Burr type XII Distributions. Biometrika, 64(1) , 129-134, doi:10.1093/biomet/64.1.129.
- Rubinstein, R. Y., Kroese, D. P., (2006). Simulation and the Monte Carlo method. 2nd edition, John Wiley and Sons, Inc., Hoboken, New Jersey.
- Shafiq, M., Viertl, R. (2014). Maximum likelihood estimation for Weibull distribution in case of censored fuzzy life time data. http://www.statistik.tuwien.ac.at/forschung/SM/ SM-2014-2complete.pdf. pp. 1-17.
- Shafiq, M., Atif, M., (2015). On the survival models for step-stress experiments based on fuzzy life time data. Qual Quant, doi: 10.1007/s11135-015-0295-9.
- Singh S., Singh U., Kumar M., (2016). Bayesian estimation for Poission-exponential model under Progressive type-II censoring data with Binomial removal and its application to ovarian cancer data, Communications in Statistics-Simulation and Computation, 45, pp. 3457-3475.
- Singpurwalla, N. D., Booker, J. M., (2004). Membership functions and probability measures of fuzzy sets, Journal of the American Statistical Association, 99(467), pp. 867- 877.
- Soliman, A. A., (2005). Estimation of parameters of life from progressively censored data using Burr-XII model. IEEE Trans. Reliab., 54, 34-42.
- Tadikamalla, P. R., (1980). A Look at the Burr and Related Distributions. International Statistical Review, 48(3) , pp. 337-344.
- Tanaka, H., Okuda, T., Asai, K., (1979). Fuzzy information and decision in statistical model. In: Advances in Fuzzy Sets Theory and Applications, North-Holland, Amsterdam, pp. 303-320.
- Tierney, L., Kadane, J. B., (1986). Accurate approximations for posterior moments and marginal densities. Journal of the American Statistical Association, 81, pp. 82-86.
- Viertl, R., (2011). Statistical Methods for Fuzzy Data, Wiley, Chichester.
- Viertl, R., (2006). Univariate statistical analysis with fuzzy data. Computational Statistics & Data Analysis, 55(1), pp. 133-147.
- Wingo, D. R., (1983). Maximum likelihood methods for fitting the Burr Type XII distribution to life test data. Biom J., 25, pp. 77-84.
- Wingo, D. R., (1993). Maximum likelihood methods for fitting the Burr type XII distribution to multiply(progressively) censored life test data. Metrika, 40, pp. 203-210.
- Wu, H. C. (2004), Fuzzy Bayesian estimation on lifetime data. Computational Statistics, 19, pp. 613-633.
- Wu, J. W., Yu, H. Y., (2005). Statistical inference about the shape parameter of the Burr type XII distribution under the failure-censored sampling plan. Applied Math. Computat., 163, pp. 443-482.
- Xiuchun, L., S. Yimin, W. Jieqiong, C., Jian, (2007). Empirical Bayes estimators of reliability performances using LINEX loss under progressively Type-II censored samples. Math. Comput. Simulat., 73, pp. 320-326.
- Zadeh, L., (1965). Fuzzy sets Information and Control, 8(3), pp. 338-353.
- Zadeh, L. A., (1968). Probability measures of fuzzy events. Journal of Mathematical Analysis and Applications, 10, pp. 421-427.
- Zarei, R., Amini, M., Taheri, S.M., Rezaei, A.H., (2012). Bayesian estimation based on vague lifetime data. Soft Computing, 16, pp. 165-174.
- Cited by
- ISSN
- 1234-7655
- Language
- eng
- URI / DOI
- http://dx.doi.org/10.59170/stattrans-2024-008