An exceedance model based on bivariate order statistics

Year 2021, , 785 - 795, 31.12.2021

Ayşegül Erem

https://doi.org/10.31801/cfsuasmas.816462

Cited By: 2

Abstract

In hydrologic risk analysis, the use of exceedance statistics are very important. In this sense, we construct a random threshold model based on bivariate order statistics. The exact distribution of exceedance statistics is calculated under some well-known copulas such as independent and Farlie-Gumbel-Morgenstern (FGM) copulas. Furthermore, numerical results are provided for expected value and variance of exceedance statistics under independent and Farlie-Gumbel-Morgenstern copulas. The application of the model in hydrology is also discussed.

Keywords

Exceedance Probability, bivariate order statistics, bivariate distributions, random threshold model, hydrologic risk

References

• Ariff, N. M., Jemain, A. A., Ibrahim, K., Zin, W. W., IDF relationships using bivariate copula for storm events in Peninsular Malaysia. Journal of Hydrology, 470 (2012), 158-171. https://doi.org/10.1016/j.jhydrol.2012.08.045.
• Bairamov, I.G., Some distribution free properties of statistics based on record values and characterizations of the distributions through a record. J. Appl. Statist. Sci., 5 (1997), 17-25.
• Barakat, H. M., On moments of bivariate order statistics. Annals of the Institute of Statistical Mathematics, 51(2) (1999), 351-358.
• Bayramoglu, I., Eryilmaz, S., Order statistics of dependent sequences consisting of two different sets of exchangeable variables. Journal of Computational and Applied Mathematics, 286 (2015), 1-6. https://doi.org/10.1016/j.cam.2015.02.045.
• Bayramoglu, I., Giner, G., Order statistics and exceedances for some models of INID random variables, Brazilian Journal of Probability and Statistics, 28 (4) (2014), 492-514. https://doi.org/10.1214/13-BJPS221.
• Bairamov, I. G., Petunin, Y. I., Statistical tests based on training samples, Cybernetics and Systems Analysis, 27 (3) (1991), 408-413.
• Erem, A., Bayramoglu, I., Exact and asymptotic distributions of exceedance statistics for bivariate random sequences. Statistics & Probability Letters, 125 (2017), 181-188. https://doi.org/10.1016/j.spl.2017.02.012.
• Erem, A., Bivariate two sample test based on exceedance statistics. Communications in Statistics-Simulation and Computation, 49(9) (2020), 2389-2401. https://doi.org/10.1016/j.spl.2017.02.012
• Erem, A., Bayramoglu, I., Bivariate general random threshold models and exceedance statistics. TWMS Journal of Pure and Applied Mathematics, 11(2) (2020), 189-203.
• Eryılmaz, S., Bairamov, I. G., On a new sample rank of an order statistics and its concomitant. Statistics & Probability Letters, 63(2) (2003), 123-131. https://doi.org/10.1016/S0167- 7152(03)00059-2
• Eryilmaz, S., The longest run statistic associated with exchangeable binary variables. Turkish Journal of Engineering and Environmental Sciences, 29(2) (2005), 105-112.
• Eryilmaz, S., Gebizlioglu, O. L., Tank, F., Modeling of claim exceedances over random thresholds for related insurance portfolios. Insurance: Mathematics and Economics, 49(3) (2011), 496-500. https://doi.org/10.1016/j.insmatheco.2011.08.009
• Huang, Y., Liang, Z., Hu, Y., Li, B., Wang, J., Theoretical derivation for the exceedance probability of corresponding ‡ood volume of the equivalent frequency regional composition method in hydrology. Hydrology Research, 51(6) (2020), 1274-1292. https://doi.org/10.2166/nh.2020.027
• Kemalbay, G., Bayramoglu, I., Joint distribution of new sample rank of bivariate order statistics. Journal of Applied Statistics, 42(10) (2015), 2280-2289. https://doi.org/10.1080/02664763.2015.1023705
• Nathan, R., Jordan, P., Scorah, M., Lang, S., Kuczera, G., Schaefer, M., Weinmann, E. Estimating the exceedance probability of extreme rainfalls up to the probable maximum precipitation, Journal of Hydrology, 543 (2016), 706-720. https://doi.org/10.1016/j.jhydrol.2016.10.044.
• Papaioannou, G., Kohnová, S., Bacigál, T., Szolgay, J., Hlavcová, K., Loukas, A., Joint modelling of ‡ood peaks and volumes: A copula application for the Danube River. Journal of Hydrology and Hydromechanics, 64(4) (2016), 382-392. https://doi.org/10.1515/johh-2016-0049
• Stoimenova, E., The power of exceedance-type tests under lehmann alternatives. Communications in Statistics-Theory and Methods, 40(4) (2011), 731-744. https://doi.org/10.1080/03610920903453475
• Stoimenova, E., Balakrishnan, N., A class of exceedance-type statistics for the two-sample problem. Journal of Statistical Planning and Inference, 141 (9) (2011), 3244-3255. https://doi.org/10.1080/03610920903453475
• Thompson, J. A., Bissett, W. T., Sweeney, A. M., Evaluating geostatistical modeling of exceedance probability as the first step in disease cluster investigations: very low birth weights near toxic Texas sites. Environmental Health, 13(1) (2014), 1-6.
• Yue, S., Ouarda, T. B., Bobée, B., A review of bivariate gamma distributions for hydrological application. Journal of Hydrology, 246(1-4) (2001), 1-18. https://doi.org/10.1016/S0022- 1694(01)00374-2
• Wesolowski, J., Ahsanullah, M., Distributional properties of exceedance statistics. Ann. Inst. Statist. Math., 50 (1998), 543-565.