Estimating the mean of heavy-tailed distributions
For i.i.d. observations \(X=(X_1,\dots,X_n)\) with CDF \[ F(x)=1-cx^{-1/\xi}(1+x^{-\delta}L(x)) \] (\(L\) being a slowly varying function) the problem of mean \({\mathbf E}X_1\) estimation is considered in the case \(\xi\in(1/2,1)\). (For \(\xi\in (0,1/2)\) the sample mean is an asymptotically normal estimate of \({\mathbf E}X_1\), for \(\xi>1\) the mean doesn't exist). The author considers an estimate \[ \widehat E=\int_0^{u_n}x\,dF_n(x)+\int_{u_n}^\infty x\,d\widehat F(x), \] where \(F_n(x)\) is the empirical CDF evaluated by \(X\) and \(\widehat F\) is an estimate of the tail of \(F\). The tail estimation based on the Pareto approximation leads to the estimator \[ \widehat M= n^{-1} \sum_{i=1}^nX_i{\mathbf 1}_{\{X_i\leq u\}} +\widehat p_n(u_n+\widehat\beta_n/(1-\widehat\xi_n)), \] where \(\widehat p=1-F_n(u_n)\), \(\widehat\xi_n\) is an estimate of \(\xi\) and \(\widehat\beta_n\) is an estimate of \(u_n\xi\) (possibly different from \(u_n\widehat\xi_n\)). The asymptotic normality of \(\widehat M\) is demonstrated. Results of simulations are presented. Applications to telecommunication data are considered.
scientific article; zbMATH DE number 2169511