Estimation of R= P [Y < X] for Two-parameter Burr Type XII Distribution

In this article, we consider the estimation of P[Y < X], when strength, X and stress, Y are two independent variables of Burr Type XII distribution. The MLE of the R based on one simple iterative procedure is obtained. Assuming that the common parameter is known, the maximum likelihood estimator, uniformly minimum variance unbiased estimator and Bayes estimator of P[Y < X] are discussed. The exact confidence interval of the R is also obtained. Monte Carlo simulations are performed to compare the different proposed methods.

• H.Panahi
• S.Asadi

• Stress-Strength model
• Maximum likelihood estimator
• Bayes estimator
• Burr type XII distribution.

[1] I.W. Burr, "Cumulative frequency distribution," Annals of Mathematical
Statistics vol. 13, 215-232, 1942.
[2] H. Panahi, and S. Asadi, "Burr Type XII Distribution: Different Method
of Estimations," The Tenth Islamic Countries Conference on Statistical
Sciences The American University, Egypt, 2009.
[3] I.G. Surles, and W.J. Padgett, "Inference for reliability and stressstrength
for a scaled Burr Type X distribution," Lifetime Data Analysis
vol. 7, [102] 187-200, 2001.
[4] M.Z. Raqab, and D. Kundu, "Comparison of Different Estimators of P[Y
< X] for a Scaled Burr Type X Distribution,- Communication in
Statistics-Computations and Simulations vol. 34(2), 465-483, 2005.
[5] Q. Shao, "Notes on maximum likelihood estimation for the threeparameter
Burr XII distribution," Computational Statistics & Data
Analysis vol. 45, 675 - 687, 2004.
[6] D. Moore, and A.S. Papadopoulos, ÔÇÿThe Burr type XII distribution as a
failure model under various loss functions,- Microelectronics Reliability
40, 2117-2122, 2000.
[7] R.N. Rodriguez, "guide to Burr Type XII distributions," Biometrika vol.
64,129-134, 1977.
[8] K.E. Ahmad, M.E. Fakhry, and Z.F. Jaheen, "Empirical Bayes
estimation of P(Y < X) and characterization of Burr-type X model,"
Journal of Statistical Planning and Inference vol. 64, 297-308, 1997.
[9] J.G. Surles, and W.J. Padgett, "Some properties of a scaled Burr type X
distribution," Journal of Statistical Planning and Inference vol. 128,
Issue 1, 271-280, 2005.
[10] A.M. Abd-Elfattah, and R.M. Mandouh, ÔÇÿEstimation of Pr{Y<X} in
Lomax case," The 39th Annual Conference on Statistics, Computer
Science and Operation Research, ISSR, Cairo University, Egypt, part 1,
156-166, 2004.
[11] D. Kundu, and R.D. Gupta, "Estimation of P(Y<X) for Generalized
Exponential Distributions," Metrika 61(3), 291-308, 2005.
[12] H. Tong, "A note on the estimation of P(Y<X) in the exponential case,"
Technometrics vol. 16, 625, 1974. (Errara, vol. 17, 395, 1975.)
[13] H. Tong, "On the estimation of P(Y<X) for exponential families," IEE
Transactions on Reliability vol. 26, 54-56, 1977.