Estimation of \(R=P[Y < X]\) for Burr type XII distribution based on ranked set sampling


  • Amal Hassan Institute of Statistical Studies and Research, Cairo University
  • Salwa Assar Institute of Statistical Studies and Research, Cairo University
  • Marwa Yahia Institute of Statistical Studies and Research, Cairo University





Ranked set sampling (RSS) is a statistical technique for data collection that generally leads to more efficient estimators than competitors based on simple random sample (SRS). In the current paper, the estimation of R=P[Y<X] when Y and X are two independent Burr type XII distribution with common known shape parameter c will be considered. Maximum likelihood estimator is proposed to estimate R based on ranked set sampling data. These estimators will be compared in terms of their biases, mean square errors and efficiencies with known estimators based on SRS data. It is shown that the estimators based on RSS are more efficient than the corresponding SRS. The results are illustrated using simulated data.

Keywords: Burr Type XII, Stress Strength Model, Ranked Set Sampling, Method of Maximum Likelihood.


A. M. Awad, M. M. Azzam, and M. A. Hamdan. Some inference results on P (Y < X) in the bivariate exponential model, Communications in Statistics Theory and Methods, 10 (1981), 25152525.

G.A. McIntyre. A method for unbiased selective sampling, using ranked sets. Australia Journal of Agaric, 3 (1952), 385-390.

H.A. Muttlak, W.A. Abu-Dayyah, M.F. Saleh and E.Al-Sawi. Estimating P(X

H. Panahi and S. Asadi. Estimation of P(X

I.W. Burr. Cumulative frequency distribution. Annals of Mathematical Statistics, 13 (1942), 215-232.

J. D. Church and B. Harris. The estimation of reliability from stress strength relationships. Technimetrics, 12 (1970), 49-54.

K.Takahasi and K.Wakimoto. On unbiased estimates of the population mean based on the sample stratified by means of ordering. Annals of the Institute of Statistical Mathematics, 20 (1968), 1-31.

M. A. Hussian. Estimation of stress-strength model for generalized inverted exponential distribution using ranked set sampling. International Journal of Advance in Engineering & Technology. 6(6) (2014), 2354-2362.

S.Kotz, Y.Lumelskii, and M. Pensky. The stress-strength model and its generalizations, World Scientific Publishing, (2003), ISBN 981-238-057-4.

S.Sengupta and S. Mukhuti. Unbiased estimation of P(X

T.R. Dell and J. L. Clutter. Ranked set sampling with order statistics background. Biometrics, 28 (1972), 545-553.

W. Abu-Dieh, A.Dorvlo and O. AL-Saidy. Estimation of reliability in case of bivariate normal distribution using ranked set sampling with concomitant variable. International Journal of the Computer, the internet and management, 19 (2011), 161-164.

W. H. Shen. On estimation of a lognormal mean using a ranked set sample. The Indian Journal of Statistics, 3 (1994), 323-333.

X. Dong, L. Zhang and F.Li. Estimation of reliability for exponential distributions using ranked set sampling with unequal samples. Quality Technology and Quantitative Management, 10 (2013), 319-328.

Z.W. Brinbaum. On a use of the Mann-Whitney statistics. Proceedings of the Third Berkeley Symposium Math. Statistics Probability, 1 (1956), 13-17.

View Full Article: