## On tail index estimation using dependent data.(English)Zbl 0738.62026

Assume that $$X_ 1,X_ 2,\dots$$ is a sequence of dependent random variables with the same marginal distribution function $$F$$, where $$1-F$$ is regularly varying at $$\infty$$, that is, there exists an $$\alpha>0$$ such that $\{1-F(tx)\}/\{1-F(x)\}\to t^{-\alpha}\hbox { as } x\to\infty\hbox{ for all } t>0.$ The author considers the estimation problem of $$\alpha$$ based on $$X_ 1,\dots,X_ n$$, where $$-\alpha$$ is called the regular variation index of $$1-F$$. The Hill estimator $$H_ n$$ of $$\alpha^{-1}$$ is defined by $m^{-1}\sum_{j=1}^ m\log X_{(j)}-\log X_{(m+1)},$ where, for $$j=1,\dots,n$$, $$X_{(j)}$$ denotes the $$j$$th largest value of $$X_ 1,\dots,X_ n$$. The consistency and asymptotic normality of $$H_ n$$ is also obtained. The results are specified to sequences $$\{X_ i\}$$ which are strictly stationary and satisfy a certain mixing condition.

### MSC:

 62F12 Asymptotic properties of parametric estimators 62F10 Point estimation 62E20 Asymptotic distribution theory in statistics 62G05 Nonparametric estimation
Full Text: