The Heavy-Tailed Gleser Model: Properties, Estimation, and Applications

Abstract

In actuarial statistics, distributions with heavy tails are of great interest to actuaries, as they represent a better description of risk exposure through a type of indicator with a certain probability. These risk indicators are used to determine companies’ exposure to a particular risk. In this paper, we present a distribution with heavy right tail, studying its properties and the behaviour of the tail. We estimate the parameters using the maximum likelihood method and evaluate the performance of these estimators using Monte Carlo. We analyse one set of simulated data and another set of real data, showing that the distribution studied can be used to model income data.

Randomly Stopped Sums, Minima and Maxima for Heavy-tailed and Light-Tailed Distributions

“…Heavy-tailed distributions applied to financial losses and stochastic returns are described and discussed in the articles [43][44][45]. The influence of heavy-tailed distributions on actuarial statistics is examined in works [46,47]. The performance of heavy-tailed distributions in social and medical research is discussed in the papers [48,49].…”

Randomly Stopped Sums, Minima and Maxima for Heavy-Tailed and Light-Tailed Distributions

“…Heavy-tailed distributions applied to financial losses and stochastic returns are described and discussed in [50][51][52]. The influence of heavy-tailed distributions on actuarial statistics is examined in [53][54][55][56]. The performance of heavy-tailed distributions in social and medical research is discussed in [57,58].…”

Scale Mixture of Gleser Distribution with an Application to Insurance Data

“…The expression on the left side is equal to zero if r < α and does not exist if r ≥ α; the integral on the right side was analyzed in Proposition 1 (e) of the work of Olmos et al [18] and does not converge. Therefore, the complete integral diverges, and this means that no rth moments exist for the random variable Z∼SMG(1, α).…”

“…The beta function is the normalization constant of the beta distribution, i.e., we say that the random variable Y has a beta distribution with parameters a and b if its pdf is given by The Gleser (G) distribution was introduced by Gleser [17] and studied recently by Olmos et al [18]; we say that a random variable X has a G distribution if its pdf is given by…”