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Bounds for the bias of the empirical CTE

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  • Russo, Ralph P.
  • Shyamalkumar, Nariankadu D.
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    Abstract

    The Conditional Tail Expectation (CTE) is gaining an increasing level of attention as a measure of risk. It is known that nonparametric unbiased estimators of the CTE do not exist, and that , the empirical [alpha]-level CTE (the average of the n(1-[alpha]) largest order statistics in a random sample of size n), is negatively biased. In this article, we show that increasing convex order among distributions is preserved by . From this result it is possible to identify the specific distributions, within some large classes of distributions, that maximize the bias of . This in turn leads to best possible bounds on the bias under various sets of conditions on the sampling distribution F. In particular, we show that when the [alpha]-level quantile is an isolated point in the support of a non-degenerate distribution (for example, a lattice distribution) then the bias is either of the order or vanishes exponentially fast. This is intriguing as the bias of vanishes at the in-between rate of 1/n when F possesses a positive derivative at the [alpha]th quantile.

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    Bibliographic Info

    Article provided by Elsevier in its journal Insurance: Mathematics and Economics.

    Volume (Year): 47 (2010)
    Issue (Month): 3 (December)
    Pages: 352-357

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    Handle: RePEc:eee:insuma:v:47:y:2010:i:3:p:352-357

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    Web page: http://www.elsevier.com/locate/inca/505554

    Related research

    Keywords: Conditional tail expectation Tail VaR TVaR Empirical CTE;

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    1. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 9(3), pages 203-228.
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