The use of heavy-tailed distributions in actuarial science and statistics

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The main objective of the proposed research consists of using heavy-tailed distributions in actuarial science and statistics.  Statistical modeling is usually done using the normal distribution.  It is then assumed that the probability that extreme events occur is small, since the tails of the normal density are not sufficiently heavy.  Heavy-tailed distributions are useful for modeling data sets containing extreme observations, such as large claims in insurance applications.  They are also useful to construct Bayesian statistical inference methods which are robust to the presence of outliers or conflicting information.  This proposal is divided in two parts. The first part concerns statistical modeling using heavy-tailed densities, in particular the family of generalized exponential power (GEP) densities.  The GEP density has a large range of tail behaviour: exponential, polynomial and logarithmic, which is useful to model heavy or even super-heavy tails. The second part of the program concerns robustness in Bayesian inference using heavy-tailed distributions.  Conditions on the tails of the prior and the likelihood are established in order to construct robust inference, which means that the posterior density is not corrupted by conflicting information or outliers.  Conditions for robustness concern mainly the tail behaviour of the prior and the likelihood, in particular it is shown that heavy-tailed distributions are needed. This research program can find many applications in statistics, actuarial science, finance, insurance and banking industries.  For example, the presence of extreme events in finance and actuarial science such as a market crash or large insurance claims requires statistical methods using heavy-tailed distributions.