``Asymptotically unbiased'' estimators of the tail index based on external estimation of the second order parameter (Q1424663)

Estimators of the tail index \(\gamma\) of a CDF \(F\) by i.i.d. samples \(X=(X_1,\dots,X_n)\) are investigated in the case were the second order condition \[ \lim_{t\to\infty} (\ln U(tx)-\ln U(t)-\gamma\ln x)/ D\gamma t^\rho =(x^\rho-1)/ \rho \quad \forall x>0 \] holds for \(U(t)=F^{-1}(1-1/t)\). (Recall that \(\gamma=\lim_{t\to\infty}(\ln U(tx)-\ln U(t))/\ln x\)). The authors consider the approximate maximum likelihood estimator \((\widehat\gamma,\widehat D,\widehat\rho)\) for \((\gamma,D,\rho)\) based on \(k\) maximal order statistics of \(X\). Asymptotics of this estimator are investigated. The estimator \(\hat \gamma\) is compared with other estimators for \(\gamma\) via simulations.