Limit theorems for empirical processes of cluster functionals
By H. Drees, Holger Rootz'en. 2009
78 citations recorded by the source corpus.
Let (X-n, i) 1 <= i <= n,m is an element of N be a triangular array of row-wise stationary R-d-valued random variables. We use a "blocks method" to define clusters of extreme values: the rows of (X-n, i) are divided into m(n) blocks (Y-n, j), and if a block contains at least one extreme value, the block is considered to contain a cluster. The cluster starts at the first extreme value in the block and ends at the last one. The main results are uniform central limit theorems for empirical processes Z(n)(f) := 1/root nv(n) Sigma(mn)(j=1) (f(Y-n,Y- j) - Ef(Y-n,Y- j)), for v(n) = P{X-n,X- i not equal 0} and f belonging to classes of cluster functionals, that is, functions of the blocks Y-n,Y- j which only depend on the cluster values and which are equal to 0 if Y-n,Y- j does not contain a cluster. Conditions for finite-dimensional convergence include beta-mixing, suitable Lindeberg conditions and convergence of covariances. To obtain full uniform convergence, we use either "bracketing entropy" or bounds on covering numbers with respect to a random semi-metric. The latter makes it possible to bring the powerful Vapnik-Cervonenkis theory to bear. Applications include multivariate tail empirical processes and empirical processes of cluster values and of order statistics in clusters. Although our main field of applications is the analysis of extreme values, the theory can be applied more generally to rare events occurring, for example, in nonparametric curve estimation.
Subfields: Applied Mathematics, Artificial Intelligence, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/9f1f3baf9334aa33fc13705a054f3eb9726dccb1
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In Applied Mathematics
In Artificial Intelligence
In Mathematical Physics
Citations in the corpus
- A Horse Race between the Block Maxima Method and the Peak–over–Threshold Approach (2018, methodology): “Under suitable conditions on the serial dependence, estimators based on the POT approach are still consistent and asymptotically normal, though with a possibly different asymptotic variance (this can for instance be deduced from Drees and Rootzén, 2010).”
- Statistics of heteroscedastic extremes (2016, unknown): “Hsing (1991), Drees (2000) and recently Drees and Rootz´en (2010), further develop statistical tools to handle extreme events for weakly dependent observations.”