Statistics of Extremes: Animation 4

An animation from the 2015 review by A.C. Davison and R. Huser, "Statistics of Extremes," from the Annual Review of Statistics and Its Application.

Illustration of the estimation of low-probability events. The upper panels display asymptotically dependent (left) and asymptotically independent (right) data, along with a target extreme region A. The numbers correspond to the true probability p that a point lies in A (black), its naive empirical estimate pˆ (red ), its estimate pˆ _D under asymptotic dependence using Equation 12 (blue) and its estimate pˆ _I under asymptotic independence using Equation 23 (purple). For pˆ _D and pˆ _I , extrapolation is based on the empirical estimate at the 0.95 level (dashed blue lines), and pˆ _I uses an estimate ηˆ of the coefficient of tail dependence proposed in an article by Ledford & Tawn (1996). The bottom panels show these probabilities as a function of the threshold (i.e., the x-coordinate of the lower left corner of A).