Modelling extremal events: for insurance and finance
Modelling extremal events: for insurance and finance
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This contribution aims at studying the behavior of the classical sample moment estimator, S(n,q)=@?"k"="1^nX"k^q/n, as a function of the number of available samples n, in the case where the random variables X are positive, have finite moments at all orders and are naturally of the form X=expY with the tail of Y behaving like e^-^y^^^@r. This class of laws encompasses and generalizes the classical example of the log-normal law. This form is motivated by a number of applications stemming from modern statistical physics or multifractal analysis. Borrowing heuristic and analytical results from the analysis of the Random Energy Model in statistical physics, a critical moment q"c(n) is defined as the largest statistical order q up to which the sample mean estimator S(n,q) correctly accounts for the ensemble average EX^q, for a given n. A practical estimator for the critical moment q"c(n) is then proposed. Its statistical performance are studied analytically and illustrated numerically in the case of i.i.d. samples. A simple modification is proposed to explicitly account for correlation among the observed samples. Estimation performance are then carefully evaluated by means of Monte-Carlo simulations in the practical case of correlated time series.