Berry-Esseen bounds for compound-Poisson loss percentiles

Abstract

The Berry–Esseen (BE) theorem of probability theory is employed to establish bounds on percentile estimates for compound-Poisson loss portfolios. We begin by arguing that these bounds should not be based upon the exact BE constant, but rather upon a possibly lower, asymptotic counterpart for which the Lyapunov fraction converges uniformly to zero. We use this constant to construct two bounds – one approximate, and the other exact – and then propose a simple numerical criterion for determining whether the Gaussian approximation affords sufficient accuracy for a given Poisson mean and individual-loss distribution. Applying this criterion to the cases of gamma and lognormal individual losses, we find there exists a positive lower bound for the minimum Poisson mean necessary to achieve a fixed degree of accuracy for losses generated by the ‘best-case’ individual-loss distribution. Further investigation of this ‘best case’ reveals that large minimum Poisson means (i.e. 700) are required to achieve reasonable accuracy for the 99th percentile associated with these losses. Finally, we consider how the upper BE bound of a tail percentile may be applied to a common practical problem: selecting excess-of-loss reinsurance retentions.

Keywords:

• Berry–Esseen theorem
• compound-Poisson sums
• percentile estimation
• risk theory
• reinsurance retention

Acknowledgements

The authors thank the anonymous referee for his/her very helpful comments.

Notes

No potential conflict of interest was reported by the authors.

1 Note that the phrase ‘non-standardized sum’, wherever it occurs, actually means ‘not necessarily standardized sum’, since standardized sums are permitted as special cases.

2 The distribution is commonly characterized by its probability density function (PDF), .

3 The distribution is commonly characterized by its PDF, .

Additional information

Funding

The fourth author acknowledges financial support from the National Science Foundation of China (NSFC) [grant number 71301161 and grant number 71532013].