JDS Journal of Data Science 1683-86021680-743X1680-743X School of Statistics, Renmin University of China JDS1067 10.6339/22-JDS1067 Statistical Data Science EVIboost for the Estimation of Extreme Value Index Under Heterogeneous Extremes WangJiaxi1 HouYanxiyxhou@fudan.edu.cn1∗ LiXingchi2 WangTiandong3 School of Data Science, Fudan University, China Department of Statistics, Texas A&M University, USA Shanghai Center for Mathematical Sciences, Fudan University, China Corresponding author. Email: yxhou@fudan.edu.cn. 20233102022214638657 Supplementary Material

The following files are included in the supplementary material: (1) Programs for modeling TIR and EVIboost; (2) Code files for simulation study, along with the detailed experiment results; (3) Code and data files for financial data analysis via EVIboost; (4) Simulation results on computational time of the EVIboost model.

18620221692022 2023 The Author(s). Published by the School of Statistics and the Center for Applied Statistics, Renmin University of China.2023 Open access article under the CC BY license.

Modeling heterogeneity on heavy-tailed distributions under a regression framework is challenging, yet classical statistical methodologies usually place conditions on the distribution models to facilitate the learning procedure. However, these conditions will likely overlook the complex dependence structure between the heaviness of tails and the covariates. Moreover, data sparsity on tail regions makes the inference method less stable, leading to biased estimates for extreme-related quantities. This paper proposes a gradient boosting algorithm to estimate a functional extreme value index with heterogeneous extremes. Our proposed algorithm is a data-driven procedure capturing complex and dynamic structures in tail distributions. We also conduct extensive simulation studies to show the prediction accuracy of the proposed algorithm. In addition, we apply our method to a real-world data set to illustrate the state-dependent and time-varying properties of heavy-tail phenomena in the financial industry.

 gradient boosting heterogeneous extremes Pareto model tail estimation tree-based method National Natural Science Foundation of China72171055Natural Science Foundation of Shanghai20ZR1403900Yanxi Hou’s research was partly supported by the National Natural Science Foundation of China Grant 72171055 and the Natural Science Foundation of Shanghai Grant 20ZR1403900. References Adrian T, Brunnermeier MK (2016). Covar. The American Economic Review, 106: 1705–1741. Breiman L, Friedman J, Stone C, Olshen R (1984). Classification and Regression Trees. CRC Press, Abingdon, United Kingdom. Clauset A, Shalizi CR, Newman MEJ (2009). Power-law distributions in empirical data. SIAM Review, 51(4): 661–703. De Haan L, Ferreira A (2006). Extreme Value Theory: An Introduction (Vol. 21). Springer, New York. Dekkers A, De Haan L (1989). On the estimation of the extreme-value index and large quantile estimation. The Annals of Statistics, 17(4): 1795–1832. DiCiccio TJ, Efron B (1996). Bootstrap confidence intervals. Statistical Science, 11(3): 189–228. Einmahl J, De Haan L, Zhou C (2016). Statistics of heteroscedastic extremes. Journal of the Royal Statistical Society, Series B, Statistical Methodology, 78(1): 31–51. Friedman J (2001). Greedy function approximation: A gradient boosting machine. The Annals of Statistics, 29(6): 1189–1232. Gençay R, Selçuk F, Ulugülyaǧci A (2003). High volatility, thick tails and extreme value theory in value-at-risk estimation. Insurance. Mathematics & Economics, 33(2): 337–356. Hall P (1982). On some simple estimates of an exponent of regular variation. Journal of the Royal Statistical Society, Series B, Methodological, 44(1): 37–42. Hill B (1975). A simple general approach to inference about the tail of a distribution. The Annals of Statistics, 3(5): 1163–1174. Meinshausen N (2006). Quantile regression forests. Journal of Machine Learning Research, 7(35): 983–999. Mentch L, Hooker G (2016). Quantifying uncertainty in random forests via confidence intervals and hypothesis tests. Journal of Machine Learning Research, 17(26): 1–41. Sandri M, Zuccolotto P (2008). A bias correction algorithm for the Gini variable importance measure in classification trees. Journal of Computational and Graphical Statistics, 17(3): 611–628. Wager S, Hastie T, Hastie T (2014). Confidence intervals for random forests: The jackknife and the infinitesimal jackknife. Journal of Machine Learning Research, 15(1): 1625–1651. Wang H, Tsai C (2009). Tail index regression. Journal of the American Statistical Association, 104(487): 1233–1240. White AP, Liu WZ (1994). Technical mote: Bias in information-based measures in decision tree induction. Machine Learning, 15(3): 321–329. Xu W, Hou Y, Li D (2020). Prediction of extremal expectile based on regression models with heteroscedastic extremes. Journal of Business & Economic Statistics, 40(2): 522–536. Yang Y, Qian W, Zou H (2018). Insurance premium prediction via gradient tree-boosted Tweedie compound Poisson models. Journal of Business & Economic Statistics, 36(3): 456–470. Zhang T, Yu B (2005). Boosting with early stopping: Convergence and consistency. The Annals of Statistics, 33(4): 1538–1579. Zhao Z, Zhang Z, Chen R (2018). Modeling maxima with autoregressive conditional Fréchet model. Journal of Econometrics, 207(2): 325–351.