Abstract
AbstractIn various applications of heavy-tail modelling, the assumed Pareto behaviour is tempered ultimately in the range of the largest data. In insurance applications, claim payments are influenced by claim management and claims may, for instance, be subject to a higher level of inspection at highest damage levels leading to weaker tails than apparent from modal claims. Generalizing earlier results of Meerschaert et al. (2012) and Raschke (2020), in this paper we consider tempering of a Pareto-type distribution with a general Weibull distribution in a peaks-over-threshold approach. This requires to modulate the tempering parameters as a function of the chosen threshold. Modelling such a tempering effect is important in order to avoid overestimation of risk measures such as the value-at-risk at high quantiles. We use a pseudo maximum likelihood approach to estimate the model parameters and consider the estimation of extreme quantiles. We derive basic asymptotic results for the estimators, give illustrations with simulation experiments and apply the developed techniques to fire and liability insurance data, providing insight into the relevance of the tempering component in heavy-tail modelling.
Highlights
Probability distributions with power-law tails are extensively used in various elds of applications including insurance, nance, information technology, mining of precious stones and language studies
In the context of insurance data, Raschke [11] recently discussed the use of the more general Weibull tempering of a simple power law with
In case τ > 1 when the tempering is quite strong, the results for the proposed methods are clearly improving upon the classical estimators Hk,n and QHp,k. Note that in these cases the V aR estimates based on the MLE parameters taken at the adaptive value kshow a rather small bias, even in case of the log-normal model which is situated outside our Pareto-type model assumption
Summary
Probability distributions with power-law tails are extensively used in various elds of applications including insurance, nance, information technology, mining of precious stones and language studies (see e.g. [10] for a recent overview). In view of the general nature of the Pareto-type models (1), this approach will not be able to capture the characteristics over the whole range of the distribution but focuses rather on the largest observations above some threshold Xn−k,n If appropriate such tempered tail ts could be spliced with dierent methods to describe the data below the chosen Xn−k,n, as it was done before to obtain composed models with a Pareto or generalized Pareto tail t; see for instance Reynkens et al [12] for mixed Erlang compositions with Pareto tails, Brazauskas and Kleefeld [13] for lognormal and Weibull models spliced with Pareto tail ts, and Raschke [11] for Pareto-Pareto or cascade Pareto modelling.
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