Logarithm of ratios of two order statistics and regularly varying tails

Here we suppose that the observed random variable has cumulative distribution function F with regularly varying tail, i.e., 1 − F ∈ RV−α, α > 0. Using the results about exponential order statistics we investigate logarithms of ratios of two order statistics of a sample of independent observations of a Pareto distributed random variable with parameter α. Short explicit formulae for its mean and variance are obtained. Then we transform this function in a proper way to obtain unbiased, asymptotically efficient, and asymptotically normal estimator for α. 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.Here we suppose that the observed random variable has cumulative distribution function F with regularly varying tail, i.e., 1 − F ∈ RV−α, α > 0. Using the results about exponential order statistics we investigate logarithms of ratios of two order statistics of a sample of independent observations of a Pareto distributed random variable with parameter α. Short explicit formulae for its mean and variance are obtained. Then we transform this function in a proper way to obtain unbiased, asymptotically efficient, and asymptotically normal estimator for α. 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.