Journal of Advanced Statistics

Download PDF (233.3 KB) PP. 146 - 155 Pub. Date: September 1, 2016

DOI: 10.22606/jas.2016.13004

• R.U. Khan^*
  Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh-202002, India
• M.A. Khan
  Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh-202002, India

In this paper, simple explicit expressions and some recurrence relations satisfied by single and product moments of generalized order statistics from exponential-Weibull lifetime distribution have been obtained. These relations are deduced for moments of order statistics and upper record values. Further, conditional moments and a recurrence relation for single moments of the generalized order statistics are used to characterize this distribution and some computational works are also carried out.

Exponential-Weibull lifetime distribution, generalized order statistics, record values, order statistics, single moments, product moments, conditional moments, recurrence relations, characterization.

[1] U. Kamps, A Concept of Generalized Order Statistics, B.G. Teubner Stuttgart, Germany, 1995.

[2] U. Kamps and U. Gather, "Characteristic property of generalized order statistics for exponential distribution", Applicationes Mathematicae (Warsaw), vol. 24, no. 4, pp. 383-391, 1997.

[3] C. Keseling, "Conditional distributions of generalized order statistics and some characterizations", Metrika, vol. 49, no. 1, pp. 27-40, 1999.

[4] E. Cramer, and U. Kamps, "Relations for expectations of functions of generalized order statistics", Journal of Statistical Planning and Inference, vol. 89, no. 1-2, pp. 79-89, 2000.

[5] M. Ahsanullah, "Generalized order statistics from exponential distribution", Journal of Statistical Planning and Inference, vol. 85, no. 1-2, pp. 85-91, 2000.

[6] M. Habibullah and M. Ahsanullah, "Estimation of parameters of a Pareto distribution by generalized order statistics", Communications in Statistics—Theory and Methods, vol. 29, no.7, pp. 1597-1609, 2000.

[7] M.Z. Raqab, "Optimal prediction-intervals for the exponential distribution based on generalized order statistics", IEEE Transactions on Reliability, vol. 50, no. 1, pp. 112-115, 2001.

[8] U. Kamps and E. Cramer, "On distributions of generalized order statistics", Statistics, vol. 35, no. 3, pp. 269-280, 2001.

[9] A.A. Ahmad and A.M. Fawzy, "Recurrence relations for single moments of generalized order statistics from doubly truncated distribution", Journal of Statistical Planning and Inference, vol. 117, no. 2, pp. 241–249, 2003.

[10] M. Bieniek and D. Szynal, "Characterizations of distributions via linearity of regression of generalized order statistics", Metrika, vol. 58, no. 3, pp. 259-271, 2003.

[11] E.K. Al-Hussaini and A.A. Ahmad, "On Bayesian predictive distributions of generalized order statistics", Metrika, vol. 57, no. 2, pp. 165-176, 2003.

[12] E. Cramer, U. Kamps and C. Keseling, "Characterization via linear regression of ordered random variables: a unifying approach", Communications in Statistics—Theory and Methods, vol. 33, no. 12, pp. 2885-2911, 2004.

[13] A.H. Khan and A.A. Alzaid, "Characterization of distributions through linear regression of non-adjacent generalized order statistics", Journal of Applied Statistical Science, vol. 13, pp. 123-136, 2004.

[14] Z.F. Jaheen, "Estimation based on generalized order statistics from the Burr model", Communications in Statistics—Theory and Methods, vol. 34, no. 4, pp. 785-794, 2005.

[15] A.H. Khan, R.U. Khan and M. Yaqub, "Characterization of continuous distributions through conditional expectation of function of generalized order statistics", Journal of Applied Probability & Statistics, vol. 1, no. 1, pp.115-131, 2006.

[16] R.U. Khan, D. Kumar and H. Athar, "Moments of generalized order statistics from Erlang-truncated exponential distribution and its characterization", International Journal of Statistics and System, vol. 5, no. 4, pp.455-464, 2010.

[17] R.U. Khan and B. Zia, "Generalized order statistics from doubly truncated linear exponential distribution and a characterization", Journal of Applied Probability & Statistics, vol. 9, no. 1, pp. 53-65, 2014.

[18] U. Kamps, Characterizations of distributions by recurrence relations and identities for moments of order statistics. In: N. Balakrishnan, N. and C.R. Rao. Handbook of Statistics 16, Order Statistics: Theory & Methods, North-Holland, Amsterdam, 1998.

[19] G.M. Cordeiro, E.M.M. Ortega, and A.J. Lemonte, "The exponential-Weibull lifetime distribution", Communications in Statistics - Simulation and Computation, vol. 84, no. 12, pp. 2592-2606, 2014.

[20] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, 7th edition, Academic Press, New York, 2007.

[21] S.M. Ruiz, "An algebraic identity leading to Wilson’s theorem", The Mathematical Gazette, vol. 80, no. 489, pp. 579-582, 1996.

[22] H.A. David and H.N. Nagaraja, Order Statistics, 3rd edition, John Wiley, New York, 2003.

[23] H. Athar and H.M. Islam, "Recurrence relations for single and product moments of generalized order statistics from a general class of distribution", Metron, vol. LXII, no. 3, pp. 327-337, 2004.

[24] R.U. Khan, A. Kulshreshtha and M.A. Khan, "Relations for moments of k-th record values from exponential-Weibull lifetime distribution", Journal of the Egyptian Mathematical Society, vol. 23, no. 3, pp. 558-562, 2015.

[25] R.U. Khan, Z. Anwar and H. Athar, "Recurrence relations for single and product moments of generalized order statistics from doubly truncated Weibull distribution", The Aligarh Journal of Statistics, vol. 27, pp. 69-79, 2007.

[26] P. Pawlas and D. Szynal, "Recurrence relations for single and product moments of generalized order statistics from Pareto, generalized Pareto, and Burr distributions", Communications in Statistics—Theory and Methods, vol.30, no. 4, pp. 739-746, 2001.

[27] A.A. Ahmad, "Single and product moments of generalized order statistics from linear exponential distribution", Communications in Statistics-Theory and Methods, vol. 37, no. 8, pp. 1162-1172, 2008.

[28] J.S. Hwang and G.D. Lin, "On a generalized moments problem II", Proceedings of the American Mathematical Society, vol. 91, no. 4, pp. 577-580, 1984.