Berry-Esseen bounds for compound-Poisson loss percentiles

Frank Y. Feng, Michael R. Powers, Rui’an Xiao and Lin Zhao

Scandinavian Actuarial Journal, 2017, vol. 2017, issue 6, 519-534

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.

Date: 2017
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DOI: 10.1080/03461238.2016.1182064

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