A modified functional delta method and its application to the estimation of risk functionals

Eric Beutner*, Henryk Zähle

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review


The classical functional delta method (FDM) provides a convenient tool for deriving the asymptotic distribution of statistical functionals from the weak convergence of the respective empirical processes. However, for many interesting functionals depending on the tails of the underlying distribution this FDM cannot be applied since the method typically relies on Hadamard differentiability w.r.t. the uniform sup-norm. In this article, we present a version of the FDM which is suitable also for nonuniform sup-norms, with the outcome that the range of application of the FDM enlarges essentially. On one hand, our FDM, which we shall call the modified FDM, works for functionals that are "differentiable" in a weaker sense than Hadamard differentiability. On the other hand, it requires weak convergence of the empirical process w.r.t. a nonuniform sup-norm. The latter is not problematic since there exist strong respective results on weighted empirical processes obtained by Shorack and Wellner (1986) [25], Shao and Yu (1996) [23], Wu (2008) [32], and others. We illustrate the gain of the modified FDM by deriving the asymptotic distribution of plug-in estimates of popular risk measures that cannot be treated with the classical FDM.

Original languageEnglish
Pages (from-to)2452-2463
Number of pages12
JournalJournal of Multivariate Analysis
Issue number10
Publication statusPublished - 1 Nov 2010
Externally publishedYes


  • Censoring
  • Central limit theorem
  • Distortion risk measure
  • Functional delta method
  • Hadamard derivative
  • Mixing
  • Weighted empirical process


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