Estimation of Extreme Depth-Based Quantile Regions

Abstract

Consider the extreme quantile region induced by the half-space depth function HD of the form Q D {x 2 R d : HD.x, P/ β}, such that P Q D p for a given, very small p > 0. Since this involves extrapolation outside the data cloud, this region can hardly be estimated through a fully non-parametric procedure. Using extreme value theory we construct a natural semiparametric estimator of this quantile region and prove a refined consistency result. A simulation study clearly demonstrates the good performance of our estimator. We use the procedure for risk management by applying it to stock market returns.

Halfspace depth and floating body

“…Theorem 32 gives a probabilistic interpretation also to the boundaries of those depth regions that correspond to the extreme depth-quantiles. It is also interesting to compare Theorem 32 with the recent advances of Einmahl et al [51] and He and Einmahl [78]. There, the authors employ extreme value theory in order to estimate P δ for low values of δ reliably from the data.…”

Depth for Curve Data and Applications

“…Thanks to these theoretical developments, it became possible to extend standard univariate descriptive statistics based on ranks to analyze multivariate observations (see, e.g., Oja 1983, Liu et al 1999. New classical inferential statistical techniques using these depth measures or some refinements were also developed, such as multivariate nonparametric testing (Li & Liu 2004, Zuo & He 2006, Chenouri et al 2012, confidence regions (Yeh & Singh 1997, Lee 2012, pvalues for hypothesis testing (Liu & Singh 1997), classification (Li et al 2012, Lange et al 2014, Paindaveine & Van Bever 2015, Dutta et al 2016, regression (Rousseeuw & Hubert 1999, Hallin et al 2010) and estimation of extreme quantiles (He & Einmahl 2017). See (Mosler 2013) for a nice introduction showing the richeness and usefulness of depth techniques.…”

Testing the Multivariate Regular Variation Model

“…Our study shows that these two datasets follow the MRV model, which implies that the MRV model is indeed a realistic assumption in these applications to financial markets. Besides, it provides support for the empirical studies in Cai, Einmahl, and De Haan (2011) and He and Einmahl (2017), in which the MRV model is assumed without a formal test.…”

“…The relevance of the MRV model is among others that the multivariate outlying regions are homothetic when taking different degrees of outlyingness. This makes extrapolation from intermediately extreme events to very extreme events possible, which makes MRV a powerful model (see, e.g., He and Einmahl 2017). Characterizing extreme outlyingness is not only important to detect outliers or anomalies, but it is also reveals the joint extreme behavior of multivariate risks, which in turn can be relevant for defining stress testing scenarios.…”

“…Again, our tests are carried out by plotting the p-values against various levels of k. Our analysis for the triplet is shown in Figure 4. In He and Einmahl (2017), when estimating the left and right extreme value indices of the three series, the threshold k is chosen at 80. At k = 80, we do not reject the null that the triplet follows an MRV distribution at the 5% level by both tests.…”