TY - JOUR

T1 - Asymptotic theory for the empirical Haezendonck-Goovaerts risk measure

AU - Ahn, Jae Youn

AU - Shyamalkumar, Nariankadu D.

N1 - Funding Information:
Shyamalkumar’s work was partially supported by a Society of Actuaries’ Center of Actuarial Excellence (CAE) Research Grant. We thank the referee for his detailed comments which helped to improve the clarity of the document.

PY - 2014/3

Y1 - 2014/3

N2 - Haezendonck-Goovaerts risk measures is a recently introduced class of risk measures which includes, as its minimal member, the Tail Value-at-Risk (T-VaR)-T-VaR arguably the most popular risk measure in global insurance regulation. In applications often one has to estimate the risk measure given a random sample from an unknown distribution. The distribution could either be truly unknown or could be the distribution of a complex function of economic and idiosyncratic variables with the complexity of the function rendering indeterminable its distribution. Hence statistical procedures for the estimation of Haezendonck-Goovaerts risk measures are a key requirement for their use in practice. A natural estimator of the Haezendonck-Goovaerts risk measure is the Haezendonck-Goovaerts risk measure of the empirical distribution, but its statistical properties have not yet been explored in detail. The main goal of this article is to both establish the strong consistency of this estimator and to derive weak convergence limits for this estimator. We also conduct a simulation study to lend insight into the sample sizes required for these asymptotic limits to take hold.

AB - Haezendonck-Goovaerts risk measures is a recently introduced class of risk measures which includes, as its minimal member, the Tail Value-at-Risk (T-VaR)-T-VaR arguably the most popular risk measure in global insurance regulation. In applications often one has to estimate the risk measure given a random sample from an unknown distribution. The distribution could either be truly unknown or could be the distribution of a complex function of economic and idiosyncratic variables with the complexity of the function rendering indeterminable its distribution. Hence statistical procedures for the estimation of Haezendonck-Goovaerts risk measures are a key requirement for their use in practice. A natural estimator of the Haezendonck-Goovaerts risk measure is the Haezendonck-Goovaerts risk measure of the empirical distribution, but its statistical properties have not yet been explored in detail. The main goal of this article is to both establish the strong consistency of this estimator and to derive weak convergence limits for this estimator. We also conduct a simulation study to lend insight into the sample sizes required for these asymptotic limits to take hold.

KW - Conditional tail expectation (CTE)

KW - Empirical CTE

KW - Orlicz premium

KW - Tail value-at-Risk (T-VaR)

UR - http://www.scopus.com/inward/record.url?scp=84892857164&partnerID=8YFLogxK

U2 - 10.1016/j.insmatheco.2013.12.003

DO - 10.1016/j.insmatheco.2013.12.003

M3 - Article

AN - SCOPUS:84892857164

VL - 55

SP - 78

EP - 90

JO - Insurance: Mathematics and Economics

JF - Insurance: Mathematics and Economics

SN - 0167-6687

IS - 1

ER -