Análise de máximos de risco Brasil e resseguros: uma abordagem Bayesiana

Abstract

Extreme value theory is often used for the analysis and forecasting of meteorological data, but its application has expanded to several areas where the occurrence of extreme events causes major financial losses, such as finance, stock market and insurance. A method of parameter estimation that has been increasingly studied and has obtained good results is the Bayesian inference. Such results are due in large part to the incorporation of a priori information that tends to improve the process of inference, reducing the uncertainties regarding parameter estimates. The objective of this work is to evaluate the application of Bayesian inference, considering different structures of a priori distributions, in the adjustment of the distributions of extreme values. We also compare the results with those obtained through estimation by maximum likelihood. For this, two practical situations were analyzed: in the first one, Brazil maximum risk data per semester was analyzed, investigating the possible presence of linear trend in the data, due to the fact of being extracted from a historical series, through the generalized distribution of extreme values and Gumbel distribution; and in the second application, the generalized Pareto distribution was used, with estimation by maximum likelihood and Bayesian inference, to model the losses related to auto insurance claims of European insurance companies. We concluded that Bayesian inference provides better results than the maximum likelihood estimators when used for estimation of parameters of generalized distributions of extreme values and Gumbel for Brazil risk data and for estimation of parameters of the distribution generalized Pareto applied to reinsurance data.