An exceedance model based on bivariate order statistics

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

1. 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.
2. 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.
3. Barakat, H. M., On moments of bivariate order statistics. Annals of the Institute of Statistical Mathematics, 51(2) (1999), 351-358.
4. 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.
5. 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.
6. Bairamov, I. G., Petunin, Y. I., Statistical tests based on training samples, Cybernetics and Systems Analysis, 27 (3) (1991), 408-413.
7. 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.
8. 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

9. Erem, A., Bayramoglu, I., Bivariate general random threshold models and exceedance statistics. TWMS Journal of Pure and Applied Mathematics, 11(2) (2020), 189-203.
10. 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
11. Eryilmaz, S., The longest run statistic associated with exchangeable binary variables. Turkish Journal of Engineering and Environmental Sciences, 29(2) (2005), 105-112.
12. 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
13. 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
14. 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
15. 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.
16. 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
17. 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
18. 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
19. 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.
20. 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
21. Wesolowski, J., Ahsanullah, M., Distributional properties of exceedance statistics. Ann. Inst. Statist. Math., 50 (1998), 543-565.