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.
Bibliographical noteFunding 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.
- Conditional tail expectation (CTE)
- Empirical CTE
- Orlicz premium
- Tail value-at-Risk (T-VaR)