On Hypothesis Test For Extremal Index

Extreme Value Theory is a theory considering the behaviour of extreme values of a sequence. Before 1980s, a large number of papers appeared, mainly considered the Extreme Value Theory for indenpendent and identically distributed ( i.i.d. ) sequences. In the last two decades, Extreme Value theory for stationary time series attacted more and more interests, and varies of methods were developed to estimate the extremal index 6 for such sequences. However, few authors considered the hypothese testing problem on the extremal index. In this paper, we shortly introduce the classical extreme value theory and basic extreme value theory of stationary sequences. We investigate the inter-exceedance times of a stationary sequence, extremal indexθ<1, do not follow geometric distribution. We present a hypothesis test problem that the inter-exceedance times of a stationary sequence, extremal index 0<\, follow mixed geometric distribution. The asymptotic distribution of the test statistic and rejected domain are given. Test power is also discussed in the end. Test power of the sequences from GARCH process and ARMAX process are given via simulation.