The asymptotics of moments and moment’s derivatives for excess distribution

The functioning of modern complex systems is characterized by various types of risks. Data analysis of such systems shows that data sets have characteristic properties: heavy distribution tails. An important issue is the impact of individual extreme events on the global behavior of the entire system too. The proposed article discusses classes of distributions with heavy tails, which are important in the theory of insurance claims and reliability theory: Gnedenko-Weibull distribution; Benktander I, II; Burr XII. The moment`s asymptotics have been derived for mean excess function and excess variance function especially for the heavy-tailed distributians. The paper studies in detail the error estimate for the asymptotic expansion of the mean excess function of the Gnedenko–Weibull distribution for any values of the shape parameter. There is a significant difference in the behavior of error estimates for values of the shape parameter less than one corresponding to the heavy tail of the Gnedenko –Weibull distribution. The values of the shape parameter are found for which the decompositions are accurate in particular. That is, the expansion has finite quantity members. Asymptotic expansions of derivatives of residual moments are proved for Gnedenko-Weibull; Benktander I, II; Burr XII distributions. The description of the behavior of the system as a region of attraction of the ultimate extreme state is also considered. These results serve as a tool for the applications to risk theory, reliability and extremal event.