Alternatives to a semi-parametric estimator of parameters of rare events -- the jackknife methodology (Q5954055)

The authors consider four estimators of the tail index \(\gamma\) of a heavy tailed d.f. \(F\) by an i.i.d. sample \(X_1\),\dots,\(X_N\). Some of them are constructed using the generalized jackknife technique (GJ). In GJ a combination of two basic estimators is used to derive bias reduction effects. Denote \(M_n^{(j)}(k)=k^{-1}\sum_{i=1}^k [\ln X_{n-i+1:n}-\ln X_{n-k:n}]\) (\(X_{k:n}\) being order statistics). Then the basic estimators used in the paper are: the classical Hill estimator \(\gamma_N^{(1)}(k)=M_n^{(1)}(k)\); de Vries estimator \(\gamma_N^{(2)}(k)=M_n^{(2)}(k)/(2\gamma_N^{(1)}(k))\), and the estimator \(\gamma_N^{(3)}(k)=\sqrt{M_n^{(2)}(k)/2}\). The GJ estimators proposed by the authors are \[ \gamma_n^{G_1}(k)=2\gamma_n^{(2)}(k)-\gamma_n^{(1)}(k);\quad \gamma_n^{G_2}(k)=4\gamma_n^{(3)}(k)-3\gamma_n^{(1)}(k); \] \[ \gamma_n^{G_3}(k)=3\gamma_n^{(2)}(k)-2\gamma_n^{(3)}(k);\quad \gamma_n^{G_4}(k)=2\gamma_n^{(1)}(k/2)-\gamma_n^{(1)}(k). \] The asymptotic bias, mean square error and the asymptotic relative efficiency of these estimators are compared using the second order behavior of \(F\). The finite sample behavior is investigated via simulations. The authors' conclusion is that ``if we had to select an estimator among the ones considered our choice would be \(\gamma_n^{G_3}(k)=3\gamma_n^{(2)}(k)-2\gamma_n^{(3)}(k)\)''.