A method for estimating the power of moments

Let X be an observable random variable with unknown distribution function F(x) = P(X≤ x) , − ∞ OpenSPiltSPi xOpenSPiltSPi ∞ , and let θ=sup{r≥0:E|X|rOpenSPiltSPi∞}.We call θ the power of moments of the random variable X. Let X₁, X₂, … , X_n be a random sample of size n drawn from F(⋅). In this paper we propose the following simple point estimator of θ and investigate its asymptotic properties: θˆn=lognlogmax1≤k≤n|Xk|where log x= ln (e∨ x) , − ∞ OpenSPiltSPi xOpenSPiltSPi ∞. In particular, we show that θˆn→Pθif and only iflimx→∞xrP(|X|CloseSPigtSPix)=∞∀rCloseSPigtSPiθ. This means that, under very reasonable conditions on F(⋅) , θˆ _n is actually a consistent estimator of θ.