Goodness-of-fit Measures Based on the Mellin Transform for Beta Generalized Lifetime Data

In recent years various probability models have been proposed for describing lifetime data. Increasing model flexibility is often sought as a means to better describe asymmetric and heavy tail distributions. Such extensions were pioneered by the beta-G family. However, efficient goodness-of-fit (GoF) measures for the beta-G distributions are sought. In this paper, we combine probability weighted moments (PWMs) and the Mellin transform (MT) in order to furnish new qualitative and quantitative GoF tools for model selection within the beta-G class. We derive PWMs for the Fr\’{e}chet and Kumaraswamy distributions; and we provide expressions for the MT, and for the log-cumulants (LC) of the beta-Weibull, beta-Fr\’{e}chet, beta-Kumaraswamy, and beta-log-logistic distributions. Subsequently, we construct LC diagrams and, based on the Hotelling’s $T^2$ statistic, we derive confidence ellipses for the LCs. Finally, the proposed GoF measures are applied on five real data sets in order to demonstrate their applicability.

Extremal Index Estimation

In finance it is crucial to understand the risk of occurrence of extreme events such as currency crises or stock market crashes. It is important to model the distribution of extreme events. Extreme value theory is known to accurately estimate quantiles and tail probabilities of financial asset returns. These kinds of data are usual related to heavy tailed distributions, where a relevant parameter is the tail index. Fitting data to heavy tail distributions usually assumes independent observations. However, the most usual real market scenario describes clusters of extreme events rather than isolated records over some period of time. In that case, estimating tail probabilities includes estimating the extremal index. This chapter describes the usual extremal index estimators based in different approaches and illustrates their values for a real financial data set. Computations are provided by the use of suitable R packages.

Aggregation of Dependent Risks with Heavy-Tail Distributions

Straightforward methods to evaluate risks arising from several sources are specially difficult when risk components are dependent and, even more if that dependence is strong in the tails. We give an explicit analytical expression for the probability distribution of the sum of non-negative losses that are tail-dependent. Our model allows dependence in the extremes of the marginal beta distributions. The proposed model is flexible in the choice of the parameters in the marginal distribution. The estimation using the method of moments is possible and the calculation of risk measures is easily done with a Monte Carlo approach. An illustration on data for insurance losses is presented.