Bootstrap consistency and bias correction in the nonparametric estimation of risk measures of collective risks

Alexandra Lauer and Henryk Zähle

Insurance: Mathematics and Economics, 2017, vol. 74, issue C, 99-108

Abstract: We consider two nonparametric estimators for the risk measure of the sum of n i.i.d. individual insurance risks divided by n, where the number of historical single claims that are used for the statistical estimation is of order n. This framework matches the situation that nonlife insurance companies are faced with within the scope of premium calculation. Indeed, the risk measure of the collective risk divided by n can be seen as a suitable premium for each of the individual risks. For both estimators asymptotic normality has been obtained recently. Here we derive almost sure bootstrap consistency for both estimators, where we allow for the weighted exchangeable bootstrap and rather general law-invariant risk measures. Both estimators are subject to a relevant negative bias for small to moderate n. For one of them we investigate by means of numerical experiments the benefit of a bootstrap-based bias correction. The numerical experiments are performed for the Value at Risk and the Average Value at Risk, and the results are comparable to those of Kim and Hardy (2007) who did analogous experiments for classical nonparametric plug-in estimators. For the other estimator the benefit of a bootstrap-based bias correction can be ruled out by theoretical arguments.

Keywords: Law-invariant risk measure; Collective risk; Nonparametric estimation; Bootstrap consistency; Weighted exchangeable bootstrap; Bootstrap-based bias correction; Value at Risk; Average Value at Risk (search for similar items in EconPapers)
Date: 2017
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S016766871630395X
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:insuma:v:74:y:2017:i:c:p:99-108

DOI: 10.1016/j.insmatheco.2017.03.001

Access Statistics for this article

Insurance: Mathematics and Economics is currently edited by R. Kaas, Hansjoerg Albrecher, M. J. Goovaerts and E. S. W. Shiu

More articles in Insurance: Mathematics and Economics from Elsevier
Bibliographic data for series maintained by Catherine Liu ().