Penultimate approximation for Hill's estimator (Q2771559)

For the Hill estimator \(H_{k,n}\) of the tail index \(\gamma\) of a d.f. \(F\) the authors derive an asymptotic approximation by the gamma distribution \(\Gamma_k\) with \(k\) degrees of freedom: NEWLINE\[NEWLINE\lim_{n\to\infty}(k a(n/k))^{-1/2}\left[ \Pr\left\{\sqrt{k}(H_{k,n}\gamma^{-1}-1)\leq x\right\}-\Gamma_k(k+x\sqrt{k}) \right] =-\gamma\phi(x)(1-\rho)^{-1},NEWLINE\]NEWLINE where \(\phi\) is the standard normal PDF, and \(a(t)\) and \(\rho\) are some second order tail behavior characteristics of \(F\).