Abstract
Inferencefor R=P(Y<X) is considered when Xand Y are independently distributed as scaled Burrtype X random variables. Under this model, exact inference proceduresfor R cannot be found. Hence, based on the expectedFisher information matrix which is derived here, asymptotic inferenceprocedures for R and other general functions ofthe parameters are developed. A bootstrap method to estimatevariance for the maximum likelihood estimators is also discussed.To illustrate these techniques, an example using carbon fiberstrength data is given. Simulations to assess the effectivenessof these techniques, as well as other concerns, are presented.
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References
K. E. Ahmad, M. E. Fakhry, and Z. F. Jaheen, “Empirical Bayes estimation of P(Y < X) and characterizations of the Burr-type X model,” Journal of Statistical Planning and Inference vol. 64 pp. 297–308, 1997.
M. G. Bader and A. M. Priest, “Statistical aspects of fibre and bundle strength in hybrid composites,” in Progress in Science and Engineering Composites, (T. Hayashi, K. Kawata, and S. Umekawa, eds.), vol. ICCM-IV, pp. 1129–1136, Tokyo, 1982.
I. W. Burr, “Cumulative frequency functions,” Annals of Mathematical Statistics vol. 13 pp. 215–222, 1942.
B. Efron, The Jackknife, the Bootstrap and Other Resampling Plans, Society for Industrial and Applied Mathematics: Philadelphia, PA, 1982.
B. Efron and R. J. Tibshirani, An Introduction to the Bootstrap, Chapman & Hall: New York, NY, 1993.
Z. F. Jaheen, “Bayesian approach to prediction with outliers from the Burr type X model,” Microelectronics Reliability vol. 35 pp. 45–47, 1995.
Z. F. Jaheen, “Empirical Bayes estimation for the reliability and failure rate functions of the Burr type X failure model,” Journal of Applied Statistical Science vol. 3 pp. 281–288, 1996.
G. S. Mudholkar and A. D. Hutson, “The exponentiated Weibull family: Some properties and a flood data application,” Communications in Statistics–Theory and Methods vol. 25 pp. 3059–3083, 1996.
G. S. Mudholkar and D. K. Srivastava, “Exponentiated Weibull family for analyzing bathtub failure-rate data,” IEEE Transactions on Reliability vol. 42 pp. 299–302, 1993.
G. S. Mudholkar, D. K. Srivastava, and M. Freimer, “The exponentiated Weibull family: A reanalysis of the bus-motor-failure data,” Technometrics vol. 37 pp. 436–445, 1995.
H. A. Sartawi and M. S. Abu-Salih, “Bayesian prediction bounds for the Burr type X model,” Communications in Statistics–Theory and Methods vol. 20 pp. 2307–2330, 1991.
J. G. Surles and W. J. Padgett, “Inference for P(Y < X) in the Burr type X model,” Journal of Applied Statistical Science vol. 7 pp. 225–238, 1998.
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Surles, J.G., Padgett, W.J. Inference for Reliability and Stress-Strength for a Scaled Burr Type X Distribution. Lifetime Data Anal 7, 187–200 (2001). https://doi.org/10.1023/A:1011352923990
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DOI: https://doi.org/10.1023/A:1011352923990