Estimating the distortion parameter of the proportional-hazard premium for heavy-tailed losses

The distortion parameter reflects the amount of loading in insurance premiums. A specific value of a given premium determines a value of the distortion parameter, which depends on the underlying loss distribution. Estimating the parameter, therefore, becomes a statistical inferential problem, which has been initiated by Jones and Zitikis [Jones, B.L., Zitikis, R., 2007. Risk measures, distortion parameters, and their empirical estimation. Insurance: Mathematics and Economics, 41, 279-297] in the case of the distortion premium and tackled within the framework of the central limit theorem. Heavy-tailed losses do not fall into this framework as they rely on the extreme-value theory. In this paper, we concentrate on a special but important distortion premium, called the proportional-hazard premium, and propose an estimator for its distortion parameter in the case of heavy-tailed losses. We derive an asymptotic distribution of the estimator, construct a practically implementable confidence interval for the distortion parameter, and illustrate the performance of the interval in a simulation study.

► Statistical inference for proportional-hazard transform is developed. ► A distortion parameter estimator for heavy-tailed losses is defined. ► Consistency of the estimator is proved. ► Confidence intervals for the distortion parameter are derived. ► Practical implementation of the results is discussed and illustrated.