On exponential representations of log-spacings of extreme order statistics
The authors consider log-spacings of the form \(Z_j=j(\log X_{n-j+1,n}-\log X_{n-j,n})\), where \(X_{j,n}\) are the order statistics of an i.i.d. sample with Pareto-type distribution \(F\), such that \(1-F(x)=x^{-1/\gamma}l(x)\), and \(\log(l(\lambda x)/l(x))\sim b(x)\int_1^\lambda \nu^{\rho-1}d\nu\) with \(b(x)\to 0\) as \(x\to\infty\). It is shown that \[ \sup_{1\leq j\leq k} \left| Z_j -\left( \gamma+b_{n,k} \left(j/ (k+1)\right)^{-\rho} \right)f_{j,n} - \beta_{j,n} \right| =o_p(b_{n,k}), \] as \(k,n\to\infty\), \(k/n\to0\), where \(b_{n,k}=b((n+1)/(k+1))\), and \(f_{j,n}\) are some independent standard exponential r.v.s and \( \left| \sum_{j=1}^k\beta_{j,n}/j \right| =o_p(b_{n,k}). \) This approximation is used to deduce the asymptotic normality of statistics of the form \(k^{-1} \sum_{j=1}^k K(j/(k+1))Z_j\) and for the estimation of the tail index \(\gamma\).
scientific article; zbMATH DE number 2059065