Logarithm of ratios of two order statistics and regularly varying tails

Abstract

Here we suppose that the observed random variable has cumulative distribution function $F$ with regularly varying tail, i.e. $1-F \in RV_{-\alpha}$, $\alpha > 0$. Using the results about exponential order statistics we investigate logarithms of ratios of two order statistics of a sample of independent observations on Pareto distributed random variable with parameter $\alpha$. Short explicit formulae for its mean and variance are obtained. Then we transform this function in such a way that to obtain unbiased, asymptotically efficient, and asymptotically normal estimator for $\alpha$. Finally we simulate Pareto samples and show that in the considered cases the proposed estimator outperforms the well known Hill, t-Hill, Pickands and Deckers-Einmahl-de Haan estimators.