A method for estimating the power of moments

Let X be an observable random variable with unknown distribution function [Formula: see text], [Formula: see text], and let [Formula: see text] We call θ the power of moments of the random variable X. Let [Formula: see text] be a random sample of size n drawn from [Formula: see text]. In this paper we propose the following simple point estimator of θ and investigate its asymptotic properties: [Formula: see text] where [Formula: see text], [Formula: see text]. In particular, we show that [Formula: see text] This means that, under very reasonable conditions on [Formula: see text], [Formula: see text] is actually a consistent estimator of θ.