The Bivariate Pareto Model Based on Ordered Random Variables

Devendra Kumar

doi:10.36753/mathenot.421460

EN

The Bivariate Pareto Model Based on Ordered Random Variables

Abstract

Generalized order statistics constitute a unified model for ordered random variables that includes order statistics and record values among others. In this article, bivariate Pareto distribution is considered. Some new simple explicit expressions for single and product moments of concomitants of generalized order statistics based on a random sample drown from the considered distribution are derived. Further, applications of these results is seen in establishing some well known results given separately for order statistics and record values and obtaining some new results. Finally, the means, and variances of the concomitants of order statistics and record values are computed for various values of the parameters.

Keywords

Generalized order statistics,order statistics,upper record values,bivariate Pareto distribution,concomitants

References

[1] Ahmad, A.A. and Fawzy, M., Recurrence relations for single moments of generalized order statistics from doubly truncated distribution. J. Statist. Plann. Inference, 117 (2003), 241-249.
[2] Ahmad, A.A., Relations for single and product moments of generalized order statistics from doubly truncated Burr type XII distribution. J. Egypt. Math. Soc., 15 (2007), 117-128.
[3] Ahsanullah, M., Generalized order statistics from exponential distribution. J. Statist. Plann. Inference, 85 (2000), 85-91.
[4] Arnold, B. C., Pareto Distribution. Wiley, New York, (1985).
[5] Arnold, B. C., A flexible family of Multivariate Pareto distributions. Journal of Statistical Planning and Inference, 24 (1990), 249-258.
[6] Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N., A First Course in Order Statistics. John Wiley, New York, (1992).
[7] Balakrishnan, N. and Cohen, A. C., Order Statistics and Inference: Estimation Methods. Academic Press, Boston, MA. (1991).
[8] Balakrishnan, N. and Lai, C.D., Continuous bivariate distributions, 2nd-edition, Springer, New York, (2009).
[9] Balakrishnan, N., Zhu, X. and Al-Zahrani, B., A recursive algorithm for the single and product moments of order statistics from the exponential-geometric distribution and some estimation methods. Comm. Statist. Theory Methods, 44 (2015), 3576-3598.
[10] Bieniek, M. and Szynal, D., Characterizations of distributions via linearity of regression of generalized order statistics. Metrika, 58 (2003), 259-271.

[11] Cramer, E., Kamps, U. and Keseling, C., Characterization via linear regression of ordered random variables: a unifying approach. Comm. Statist. Theory Methods, 33 (2004), 2885-2911.
[12] David, H. A. and Nagaraja, H. N., Order Statistics, third edition. John Wiley, New York, (2003).
[13] Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series, and Products, sixth edition. Academic Press, San Diego, (2000).
[14] Hutchinson, T. P. and Lai, C. D., Continuous Bivariate Distributions, Emphasising Applications. Rumsby Scientific Publishing, Adelaide, (1990).
[15] Kamps, U., A Concept of Generalized Order Statistics, B.G. Teubner Stuttgart, (1995).
[16] Kamps, U. and Cramer, E., On distributions of generalized order statistics. Statistics, 35 (2001), 269-280.
[17] Kamps, U. and Gather, U., Characteristic property of generalized order statistics for exponential distribution. Appl. Math. (Warsaw), 24 (1997), 383-391.
[18] Keseling, C., Conditional distributions of generalized order statistics and some characterizations. Metrika, 49 (1999), 27-40.
[19] Khan, R. U. and Kumar, D., On moments of lower generalized order statistics from exponentiated Pareto distribution and its characterization, Appl. Math. Sci., 4 (2010), 2711-2722.
[20] Kumar, D., Recurrence relations for single and product moments of generalized order statistics from p th order exponential distribution and its characterization. Journal of Statistical Research of Iran, 7 (2010), 101-112.
[21] Kumar, D., Generalized order statistics from Kumaraswamy distribution and its characterization. Tamsui Oxford journal of Mathematical Sciences, 27 (2011), 463-476.
[22] Kumar, D., On moments of lower generalized order statistics from exponentiated lomax distribution and characterization. American Journal of Mathematical and Management Sciences, 32 (2013), 238-256.
[23] Kumar, D., Exact moments of generalized order statistics from type II exponentiated log-logistic distribution. Hacettepe Journal of Mathematics and Statistics, 44 (2015a), 715-733.
[24] Kumar, D., Lower generalized order statistics based on inverse Burr distribution, American Journal of Mathematical and Management Sciences, 35(2015b), 15-35.
[25] Langseth, H., Bayesian networks with applications in reliability analysis, Ph.D. thesis, Norwegian University of Science and Technology, Norway, (2002).
[26] Lindley, D.V. and Singpurwalla, N.D., Multivariate distribution for the life lengths of a system sharing a common environment. Journal of Applied Probability, 23 (1986), 418-431.
[27] Mardia, K. V., Multivariate Pareto distributions. The Annals of Mathematical Statistics, 33 (1962), 1008-1015.
[28] Nayak, T., Multivariate Lomax distribution, properties and usefulness in reliability theory. Journal of Applied Probability, 24 (1987), 170-177.
[29] Pawlas, P. and Szynal, D., Recurrence relations for single and product moments of generalized order statistics from Pareto, generalized Pareto, and Burr distributions. Comm. Statist. Theory Methods, 30 (2001), 739-746.
[30] Prudnikov, A.P., Brychkov, Y.A., Marichev, I., Integral and series Vol. 3, More Special Functions, Gordon and Breach Science Publisher, New York, (1986).
[31] Sankaran, P.G. and Nair, N.U., A bivariate Pareto model and its applications to reliability, Naval Research Logistics, 40 (1993), 1013-1020.
[32] Sankaran, P. G. and Kundu, D., A bivariate Pareto model, Statistics, 48 (2014), 241-255.
[33] Sankaran, P. G., Nair, N.U. and John, P., Characterizations of a family of bivariate Pareto distributions, STATISTICA, LXXV (2015), 275-290.
[34] Srivastava, H.M., Karlsson, P.W., Multiple Gaussian Hypergeometric Series, John Wiley and Sons, New York, (1985).

Details

Primary Language

English

Subjects

-

Journal Section

Research Article

Authors

Devendra Kumar

Publication Date

October 30, 2016

Submission Date

June 7, 2016

Acceptance Date

-

Published in Issue

Year 2016 Volume: 4 Number: 2

DOI

https://doi.org/10.36753/mathenot.421460

IZ

https://izlik.org/JA38FB69PM

APA

Kumar, D. (2016). The Bivariate Pareto Model Based on Ordered Random Variables. Mathematical Sciences and Applications E-Notes, 4(2), 79-90. https://doi.org/10.36753/mathenot.421460

AMA

1.Kumar D. The Bivariate Pareto Model Based on Ordered Random Variables. Math. Sci. Appl. E-Notes. 2016;4(2):79-90. doi:10.36753/mathenot.421460

Chicago

Kumar, Devendra. 2016. “The Bivariate Pareto Model Based on Ordered Random Variables”. Mathematical Sciences and Applications E-Notes 4 (2): 79-90. https://doi.org/10.36753/mathenot.421460.

EndNote

Kumar D (October 1, 2016) The Bivariate Pareto Model Based on Ordered Random Variables. Mathematical Sciences and Applications E-Notes 4 2 79–90.

IEEE

[1]D. Kumar, “The Bivariate Pareto Model Based on Ordered Random Variables”, Math. Sci. Appl. E-Notes, vol. 4, no. 2, pp. 79–90, Oct. 2016, doi: 10.36753/mathenot.421460.

ISNAD

Kumar, Devendra. “The Bivariate Pareto Model Based on Ordered Random Variables”. Mathematical Sciences and Applications E-Notes 4/2 (October 1, 2016): 79-90. https://doi.org/10.36753/mathenot.421460.

JAMA

1.Kumar D. The Bivariate Pareto Model Based on Ordered Random Variables. Math. Sci. Appl. E-Notes. 2016;4:79–90.

MLA

Kumar, Devendra. “The Bivariate Pareto Model Based on Ordered Random Variables”. Mathematical Sciences and Applications E-Notes, vol. 4, no. 2, Oct. 2016, pp. 79-90, doi:10.36753/mathenot.421460.

Vancouver

1.Devendra Kumar. The Bivariate Pareto Model Based on Ordered Random Variables. Math. Sci. Appl. E-Notes. 2016 Oct. 1;4(2):79-90. doi:10.36753/mathenot.421460