Abstract
Integral representations of the exact distributions of order statistics are derived in a geometric way when three or four random variables depend on each other as the components of continuous ln,psymmetrically distributed random vectors do, n ∈ {3,4}, p > 0. Once the representations are implemented in a computer program, it is easy to change the density generator of the ln,p-symmetric distribution with another one for newly evaluating the distribution of interest. For two groups of stock exchange index residuals, maximum distributions are compared under dependence and independence modeling.
References
[1] Arellano-Valle, R. B. and Genton, M. G. (2007). On the exact distribution of linear combinations of order statistics from dependent random variables. J. Multivariate Anal. 98(10), 1876–1894. 10.1016/j.jmva.2006.12.003Search in Google Scholar
[2] Arellano-Valle, R. B. and Genton, M. G. (2008). On the exact distribution of themaximumof absolutely continuous dependent random variables. Stat. Probab. Lett. 78(1), 27–35. 10.1016/j.spl.2007.04.021Search in Google Scholar
[3] Arellano-Valle, R. B. and Richter, W.-D. (2012). On skewed continuous ln,p-symmetric distributions. Chil. J. Stat. 3(2), 193– 212. Search in Google Scholar
[4] Arnold, B. C., Castillo, E., and Sarabia, J. M. (2007). Variations on the classical multivariate normal theme. J. Stat. Plann. Inference 137(11), 3249–3260. 10.1016/j.jspi.2007.03.009Search in Google Scholar
[5] Batún-Cutz, J., González-Farías, G., and Richter, W.-D. (2013). Maximum distributions for l2,p-symmetric vectors are skewed l1,p-symmetric distributions. Stat. Probab. Lett. 83(10), 2260–2268. 10.1016/j.spl.2013.06.022Search in Google Scholar
[6] Box, G. E. and Tiao, G. C. (1973). Bayesian Inference in Statistical Analysis. Addison-Wesley Publishing Company, Reading, Mass. Search in Google Scholar
[7] Clauset, A., Shalizi, C. R., and Newman, M. (2009). Power-law distributions in empirical data. SIAM Rev. 51(4), 661–703. 10.1137/070710111Search in Google Scholar
[8] David, H. A. and Nagaraja, H. N. (2003). Order Statistics. Wiley, New York, 3rd edition. 10.1002/0471722162Search in Google Scholar
[9] Dietrich, T., Kalke, S., and Richter, W.-D. (2013). Stochastic representations and a geometric parametrization of the twodimensional Gaussian law. Chil. J. Stat. 4(2), 27–59. Search in Google Scholar
[10] Fang, K.-T., Kotz, S., and Ng, K.-W. (1990). Symmetric Multivariate And Related Distributions. Chapman and Hall, London. 10.1007/978-1-4899-2937-2Search in Google Scholar
[11] Gómez, E., Gómez-Villegas, M. A., and Marín, J. M. (1998). A multivariate generalization of the power exponential family of distributions. Commun. Stat. Theory Methods 27(3), 589–600. 10.1080/03610929808832115Search in Google Scholar
[12] Günzel, T., Richter, W.-D., Scheutzow, S., Schicker, K., and Venz, J. (2012). Geometric approach to the skewed normal distribution. J. Stat. Plann. Inference 142(12), 3209–3224. 10.1016/j.jspi.2012.06.009Search in Google Scholar
[13] Gupta, A. K. and Song, D. (1997). lp-norm spherical distributions. J. Stat. Plann. Inference 60(2), 241–260. 10.1016/S0378-3758(96)00129-2Search in Google Scholar
[14] Jamalizadeh, A. and Balakrishnan, N. (2010). Distributions of order statistics and linear combinations of order statistics from an elliptical distribution as mixtures of unified skew-elliptical distributions. J. Multivar. Anal. 101(6), 1412–1427. 10.1016/j.jmva.2009.12.012Search in Google Scholar
[15] Jondeau, E., Poon, S.-H., and Rockinger, M. (2007). FinancialModelingUnder Non-Gaussian Distributions. Springer, London. Search in Google Scholar
[16] Kalke, S. and Richter, W.-D. (2013). Simulation of the p-generalized Gaussian distribution. J. Statist. Comput. Simulation 83(4), 639–665. 10.1080/00949655.2011.631187Search in Google Scholar
[17] Kalke, S., Richter, W.-D., and Thauer, F. (2013). Linear combinations, products and ratios of simplicial or spherical variates. Comm. Stat. Theory Methods 42(3), 505–527. 10.1080/03610926.2011.579702Search in Google Scholar
[18] Moszynska, M. and Richter,W.-D. (2012). Reverse triangle inequality. Antinorms and semi-antinorms. Stud. Sci.Math. Hung. 49(1), 120–138. Search in Google Scholar
[19] Müller, K. and Richter,W.-D. (2015). Exact extreme value, product, and ratio distributions under non-standard assumptions. AStA Adv. Stat. Anal. 99(1), 1–30. 10.1007/s10182-014-0228-2Search in Google Scholar
[20] Nadarajah, S. (2003). The Kotz-type distribution with applications. Statistics 37(4), 341–358. 10.1080/0233188031000078060Search in Google Scholar
[21] Nadarajah, S. (2005). A generalized normal distribution. J. Appl. Stat. 32(7), 685–694. 10.1080/02664760500079464Search in Google Scholar
[22] Nadarajah, S. (2006). Acknowledgement of priority: The generalized normal distribution. J. Appl. Stat. 33(9), 1031–1032. 10.1080/02664760600938494Search in Google Scholar
[23] Osiewalski, J. and Steel, M. F. (1993). Robust Bayesian inference in lq-spherical models. Biometrika 80(2), 456–460. 10.1093/biomet/80.2.456Search in Google Scholar
[24] Richter, W.-D. (2007). Generalized spherical and simplicial coordinates. J. Math. Anal. Appl. 336(2), 1187–1202. 10.1016/j.jmaa.2007.03.047Search in Google Scholar
[25] Richter, W.-D. (2008a). On l2,p-circle numbers. Lith. Math. J. 48(2), 228–234. 10.1007/s10986-008-9002-zSearch in Google Scholar
[26] Richter, W.-D. (2008b). On the Pi-function for nonconvex l2,p-circle discs. Lith. Math. J. 48(3), 332–338. 10.1007/s10986-008-9016-6Search in Google Scholar
[27] Richter, W.-D. (2009). Continuous ln,p-symmetric distributions. Lith. Math. J. 49(1), 93–108. 10.1007/s10986-009-9030-3Search in Google Scholar
[28] Richter, W.-D. (2014a). Classes of standard Gaussian random variables and their generalizations. In Knif, J. and Pape, B., editors, Contributions to Mathematics, Statistics, Econometrics, and Finance, pages 53–69. University of Vaasa. Search in Google Scholar
[29] Richter, W.-D. (2014b). Geometric disintegration and star-shaped distributions. J. Stat. Distrib. Appl. 1:20. 10.1186/s40488-014-0020-6Search in Google Scholar
[30] Richter, W.-D. (2015a). Convex and radially concave contoured distributions. J. Probab. Stat. 2015. Article ID 165468. 10.1155/2015/165468Search in Google Scholar
[31] Richter, W.-D. (2015b). Norm contoured distributions in R2. In Lecture notes of Seminario Interdisciplinare di Matematica. Vol. XII volume 12, pages 179–199. Potenza: Seminario Interdisciplinare diMatematica (S.I.M.), University of Basilicata, Italy. Search in Google Scholar
[32] Richter, W.-D. and Venz, J. (2014). Geometric representations of multivariate skewed elliptically contoured distributions. Chil. J. Stat. 5(2), 71–90. Search in Google Scholar
[33] Sarabia, J. M. and Gómez-Déniz, E. (2008). Construction of multivariate distributions: a review of some recent results. SORT 32(1), 3–36. Search in Google Scholar
[34] Silverman, H. (1994). The value of reproving. PRIMUS 4(2), 151–154. 10.1080/10511979408965744Search in Google Scholar
[35] Subbotin, M. (1923). On the law of frequency of error. Rec. Math. Moscou 31(2), 296–301. Search in Google Scholar
[36] Theodossiou, P. (1998). Financial data and the skewed generalized t distribution. Manage. Sci. 44(12), 1650–1661. 10.1287/mnsc.44.12.1650Search in Google Scholar
© 2016 K. Müller and W.-D. Richter
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.
