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A Note on Nonparametric Estimation of the CTE

Published online by Cambridge University Press:  09 August 2013

Bangwon Ko
Affiliation:
Soongsil University, 11 Sangdo-Dong, Dongjak-Gu, Seoul, 156-743, South Korea, Telephone: 82-2-820-0447, E-Mail: bko@ssu.ac.kr
Ralph P. Russo
Affiliation:
Dept. of Statistics and Actuarial Science, The University of Iowa, 241 Schaeffer Hall, Iowa City, IA 52242, USA, E-mail: rrusso@stat.uiowa.edu
Nariankadu D. Shyamalkumar
Affiliation:
Dept. of Statistics and Actuarial Science, The University of Iowa, 241 Schaeffer Hall, Iowa City, IA 52242, USA, Telephone: (319)-335-1980, E-Mail: shyamal-kumar@uiowa.edu
Corresponding
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Abstract

The α-level Conditional Tail Expectation (CTE) of a continuous random variable X is defined as its conditional expectation given the event {X > qα } where qα represents its α-level quantile. It is well known that the empirical CTE (the average of the n (1 – α) largest order statistics in a sample of size n) is a negatively biased estimator of the CTE. This bias vanishes as the sample size increases, but in small samples can be significant. In this article it is shown that an unbiased nonparametric estimator of the CTE does not exist. In addition, the asymptotic behavior of the bias of the empirical CTE is studied, and a closed form expression for its first order term is derived. This expression facilitates the study of the behavior of the empirical CTE with respect to the underlying distribution, and suggests an alternative (to the bootstrap) approach to bias correction. The performance of the resulting estimator is assessed via simulation.

Type
Research Article
Copyright
Copyright © International Actuarial Association 2009

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