Bernoulli

• Bernoulli
• Volume 24, Number 1 (2018), 115-155.

Maximum entropy distribution of order statistics with given marginals

Abstract

We consider distributions of ordered random vectors with given one-dimensional marginal distributions. We give an elementary necessary and sufficient condition for the existence of such a distribution with finite entropy. In this case, we give explicitly the density of the unique distribution which achieves the maximal entropy and compute the value of its entropy. This density is the unique one which has a product form on its support and the given one-dimensional marginals. The proof relies on the study of copulas with given one-dimensional marginal distributions for its order statistics.

Article information

Source
Bernoulli, Volume 24, Number 1 (2018), 115-155.

Dates
Revised: February 2016
First available in Project Euclid: 27 July 2017

https://projecteuclid.org/euclid.bj/1501142438

Digital Object Identifier
doi:10.3150/16-BEJ868

Mathematical Reviews number (MathSciNet)
MR3706752

Zentralblatt MATH identifier
06778323

Citation

Butucea, Cristina; Delmas, Jean-François; Dutfoy, Anne; Fischer, Richard. Maximum entropy distribution of order statistics with given marginals. Bernoulli 24 (2018), no. 1, 115--155. doi:10.3150/16-BEJ868. https://projecteuclid.org/euclid.bj/1501142438

References

• [1] Arnold, B.C., Balakrishnan, N. and Nagaraja, H.N. (1992). A First Course in Order Statistics. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics 54. New York: Wiley.
• [2] Avérous, J., Genest, C. and Kochar, S.C. (2005). On the dependence structure of order statistics. J. Multivariate Anal. 94 159–171.
• [3] Bickel, P.J. (1967). Some contributions to the theory of order statistics. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calf., 1965/66), Vol. i: Statistics 575–591. Berkeley, CA: Univ. California Press.
• [4] Boland, P.J., Hollander, M., Joag-Dev, K. and Kochar, S. (1996). Bivariate dependence properties of order statistics. J. Multivariate Anal. 56 75–89.
• [5] Borwein, J.M., Lewis, A.S. and Nussbaum, R.D. (1994). Entropy minimization, $DAD$ problems, and doubly stochastic kernels. J. Funct. Anal. 123 264–307.
• [6] Butucea, C., Delmas, J.-F., Dutfoy, A. and Fischer, R. (2015). Maximum entropy copula with given diagonal section. J. Multivariate Anal. 137 61–81.
• [7] Butucea, C., Delmas, J.-F., Dutfoy, A. and Fischer, R. (2015). Nonparametric estimation of distributions of order statistics with application to nuclear engineering. In Safety and Reliability of Complex Engineered Systems: ESREL 2015 (L. Podofillini, B. Sudret, B. Stojadinovic, E. Zio and W. Kröger, eds.) CRC Press. URL.
• [8] David, H.A. and Nagaraja, H.N. (1970). Order Statistics. New York: Wiley.
• [9] de Melo Mendes, B.V.d.M. and Sanfins, M.A. (2007). The limiting copula of the two largest order statistics of independent and identically distributed samples. Braz. J. Probab. Stat. 21 85–101.
• [10] Dubhashi, D. and Häggström, O. (2008). A note on conditioning and stochastic domination for order statistics. J. Appl. Probab. 45 575–579.
• [11] Hu, T. and Chen, H. (2008). Dependence properties of order statistics. J. Statist. Plann. Inference 138 2214–2222.
• [12] Jaworski, P. (2009). On copulas and their diagonals. Inform. Sci. 179 2863–2871.
• [13] Jaworski, P. and Rychlik, T. (2008). On distributions of order statistics for absolutely continuous copulas with applications to reliability. Kybernetika (Prague) 44 757–776.
• [14] Kim, S.H. and David, H.A. (1990). On the dependence structure of order statistics and concomitants of order statistics. J. Statist. Plann. Inference 24 363–368.
• [15] Lebrun, R. and Dutfoy, A. (2014). Copulas for order statistics with prescribed margins. J. Multivariate Anal. 128 120–133.
• [16] Navarro, J. and Balakrishnan, N. (2010). Study of some measures of dependence between order statistics and systems. J. Multivariate Anal. 101 52–67.
• [17] Navarro, J. and Spizzichino, F. (2010). On the relationships between copulas of order statistics and marginal distributions. Statist. Probab. Lett. 80 473–479.
• [18] Rüschendorf, L. and Thomsen, W. (1993). Note on the Schrödinger equation and $I$-projections. Statist. Probab. Lett. 17 369–375.
• [19] Schmitz, V. (2004). Revealing the dependence structure between $X_{(1)}$ and $X_{(n)}$. J. Statist. Plann. Inference 123 41–47.