# zbMATH — the first resource for mathematics

Inference of $$R=P(Y<X)$$ for two-parameter Rayleigh distribution based on progressively censored samples. (English) Zbl 1440.62353
Summary: The UMVUE and maximum likelihood estimator of $$R=P(Y<X)$$ for independent progressively Type-II censored samples from two-parameter Rayleigh distributions with different scale parameters are derived. Also the exact, asymptotic and bootstrap confidence intervals for $$R$$ are evaluated. Using Gibbs sampling, the Bayes estimates and corresponding credible intervals for $$R$$ are obtained. Applying Monte Carlo simulations, we compare the performances of the different estimation methods. Finally we use of two real data sets and show the competitive performance of the presented methods.

##### MSC:
 62N02 Estimation in survival analysis and censored data 62F10 Point estimation 62F15 Bayesian inference 62F25 Parametric tolerance and confidence regions
Full Text:
##### References:
 [1] Balakrishnan, N.; Aggarwala, R., Progressive censoring: theory, methods and applications, (2000), Boston: Birkhauser, Boston [2] On a use of Mann–Whitney statistics. Proceedings of the Third Berkley Symposium in Mathematics, Statistics and Probability. Vol. 1. 1956. p. 13–17 [3] Kotz, S.; Lumelskii, Y.; Pensky, M., The stress–strength model and its generalization: theory and applications, (2003), Singapore: World Scientific, Singapore · Zbl 1017.62100 [4] Kundu, D.; Gupta, RD., Estimation of $$##?##$$ for the generalized exponential distribution, Metrika, 61, 291-308, (2005) · Zbl 1079.62032 [5] Kundu, D.; Gupta, RD., Estimation of $$##?##$$ for Weibull distribution, IEEE Trans Reliab, 55, 270-280, (2006) [6] Raqab, MZ; Kundu, D., Comparison of different estimators of $$##?##$$ for a scaled Burr Type X distribution, Commun Stat Simul Comput, 34, 465-483, (2005) · Zbl 1065.62172 [7] Krishnamoorthy, K.; Mukherjee, S.; Guo, H., Inference on reliability in two-parameter exponential stress–strength model, Metrika, 65, 261-273, (2007) · Zbl 1433.62061 [8] Raqab, MZ; Madi, MT; Kundu, D., Estimation of $$##?##$$ for the 3-parameter generalized exponential distribution, Commun Stat Theory Methods, 37, 2854-2864, (2008) · Zbl 1292.62041 [9] Kundu, D.; Raqab, MZ., Estimation of $$##?##$$ for three parameter Weibull distribution, Stat Prob Lett, 79, 1839-1846, (2009) · Zbl 1169.62012 [10] Panahi, H.; Asadi, A., Estimation of $$##?##$$ for two-parameter Burr type XII distribution, World Acad Sci Eng Technol, 73, 509-514, (2011) [11] Lio, YL; Tsai, TR., Estimation of $$##?##$$ for Burr XII distribution based on the progressively first failure-censored samples, J Appl Stat, 39, 309-322, (2012) [12] Babayi, S.; Khorram, E.; Tondro, F., Inference of $$##?##$$ for generalized logistic distribution, Statistics, 48, 862-871, (2014) · Zbl 1326.62047 [13] Chaturvedi, A.; Kumari, T., Estimation and testing procedures for the reliability functions of a general class of distributions, Commun Stat Theory Methods, 46, 11370-11382, (2017) · Zbl 1379.62021 [14] Khan, HMR; Provost, SB; Singh, A., Predictive inference from a two-parameter Rayleigh life model given a doubly censored sample, Commun Stat Theory Methods, 39, 1237-1246, (2010) · Zbl 1188.62291 [15] Dey, S.; Dey, T.; Kundu, D., Two-parameter Rayleigh distribution: different methods of estimation, Amer J Math Manage Sci, 33, 55-74, (2014) [16] Cao, JH; Cheng, K., An introduction to the reliability mathematics, (2006), Beijing: Higher Education Press, Beijing [17] Johnson, NL; Kotz, S.; Balakrishnan, N., Continuous univariate distributions, (1994), Wiley, New York · Zbl 0811.62001 [18] The jackknife, the bootstrap and other re-sampling plans. CBMSNSF Regional Conference Series in Applied Mathematics. Vol. 34. Philadelphia, PA: SIAM; 1982 [19] Hall, P., Theoretical comparison of bootstrap confidence intervals, Ann Stat, 16, 927-953, (1988) · Zbl 0663.62046 [20] Devroye, L., A simple algorithm for generating random variates with a log-concave density, Computing, 33, 247-257, (1984) · Zbl 0561.65004 [21] Geman, S.; Geman, A., Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images, IEEE Trans Pattern Anal Mach Intell, 6, 721-740, (1984) · Zbl 0573.62030 [22] Chen, MH; Shao, QM., Monte Carlo estimation of Bayesian credible and HPD intervals, J Comput Graph Stat, 8, 69-92, (1999) [23] Lindley, DV., Approximate Bayesian methods, Trab Estad, 3, 281-288, (1980) [24] Ahmad, KE; Fakhry, ME; Jaheen, ZF., Empirical Bayes estimation of $$##?##$$ and characterization of Burr-type X model, J Stat Plann Inference, 64, 297-308, (1997) · Zbl 0915.62001 [25] Statistical aspects of fiber and bundle strength in hybrid composites. In: Hayashi T, Kawata K, Umekawa S, editors. Progress in Science and Engineering Composites. Tokyo: ICCM-IV; 1982. p. 1129–1136
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.