Simulation Study of the Calibration Technique in the Extremal Index Estimation
Classical extreme value methods were first derived when the underlying process is assumed to be a sequence of independent and identically distributed random variables. However, when observations are taken along the time and/or the space, the independence
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Abstract
Classical extreme value methods were first derived when the underlying process is assumed to be a sequence of independent and identically distributed random variables. However, when observations are taken along the time and/or the space, the independence is an unrealistic assumption. A relevant parameter that arises in this situation is the extremal index, , characterizing the degree of local dependence in the extremes of a stationary series. Most of the semi-parametric estimators of this parameter show a strong dependence on the threshold un , with an increasing bias and a decreasing variance as such a threshold decreases. A procedure based on the calibration methodology is here considered as a way of controlling the bias of an estimator. Point and interval estimates for the extremal index are obtained. A simulation study has been performed to illustrate the procedure.
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Introduction
Classical extreme value theory gives conditions for the existence of normalizing sequences fan > 0g and fbn g such that for un D an x C bn , P fMn un g ! G.x/ as n ! 1, where G./ is a non-degenerate distribution function that necessarily
D.P. Gomes () J.T. Mexia CMA and Mathematics Department, Faculdade de Ciˆencias e Tecnologia, Universidade Nova de Lisboa, Lisboa, Portugal e-mail: [email protected]; [email protected] M.M. Neves CEAUL and Mathematics Department, Instituto Superior de Agronomia, Universidade T´ecnica de Lisboa, Lisboa, Portugal e-mail: [email protected] J. Lita da Silva et al. (eds.), Advances in Regression, Survival Analysis, Extreme Values, Markov Processes and Other Statistical Applications, Studies in Theoretical and Applied Statistics, DOI 10.1007/978-3-642-34904-1 40, © Springer-Verlag Berlin Heidelberg 2013
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belongs to one of the Gumbel, Fr´echet and Weibull families that are usually termed as the extreme value distributions and Mn is the maximum of a sequence of independent and identically distributed (i.i.d.) random variables. However, in real situations, extreme events often occur in clusters of large values. So, for a dependent structure, the exceedances over a high level tend to occur in clusters instead of happening in an isolated way. The characterization of extremes of stationary processes, the most natural generalization of a sequence of i.i.d random variables, appeared then. b n g the Let fXn g be a stationary sequence, where Mn is the maximum and fX c associated i.i.d. sequence, with the same marginal distribution F . Let M n be the maximum of the i.i.d. sequence. If the distribution of the maximum c M n suitably normalized by constants fan > 0g and fbn g converges to a non-degenerate law, i.e., P Œc M n an x C bn ! G.x/, where G./ is the extreme value distribution, denoted by GEV .; ı; /, then the distribution of Mn also converges with the same set of normalizing constants to G ./, where is the extremal index, see [14]. G ./ is a GEV . ; ı ; / distribution with D C ı
1 ;
ı D ı ;
D :
In a dependent setup the es
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