Smoothing the moment estimator of the extreme value parameter
Let \(\{X_n\}\) be a sequence of independent random variables whose common distribution function \(F(X)\) belongs to the domain of attraction of an extreme value distribution \(G(x)=\exp\{-(1+\gamma x)^{-1/\gamma}\}\), \(\gamma \in R^1\). This paper focused on the moment estimator \(\gamma_{k,n}\) for an extreme value parameter \(\gamma\) based on \(K\) upper order statistics, proposed by \textit{A.L.M. Dekkers, J.H.J. Einmahl} and \textit{L. de Haan} [Ann. Stat. 17, No. 4, 1833-1855 (1989; Zbl 0701.62029]. The smoothed modification \[ av\gamma_{k,n}=(1/k(1-s))\sum_{p=[ks]+1}^k \gamma_{p,n}, \] where \(s\in [0,1]\), \([x]\) entire of \(x\), is obtained through an average technique. The new estimator is based on the tail of empirical processes and this fact is utilized in investigation of its asymptotic properties. Also, \(av\gamma_{k,n}\) offers some advantages in practice; for instance, it yields a less volatile function of \(k\) and, thus, simplifies the choice of \(k\). A graphical comparison of the performance of both estimators indicates that the moment estimator is superior for \(\gamma>0\) and inferior for \(\gamma<0\). An example of application of smoothed estimation for a real data set from hydrology is presented.
scientific article; zbMATH DE number 1324027