A method for estimating the power of moments.

Let <italic>X</italic> be an observable random variable with unknown distribution function F(x)=P(X≤x)<inline-graphic></inline-graphic>, −∞<x<∞<inline-graphic></inline-graphic>, and let θ=sup{r≥0:E|X|r<∞}.<graphic></graphic> We call <italic>θ</italic> the power of moments of the random variable <italic>X</italic>. Let X1,X2,…,Xn<inline-graphic></inline-graphic> be a random sample of size <italic>n</italic> drawn from F(⋅)<inline-graphic></inline-graphic>. In this paper we propose the following simple point estimator of <italic>θ</italic> and investigate its asymptotic properties: θˆn=lognlogmax1≤k≤n|Xk|,<graphic></graphic> where logx=ln(e∨x)<inline-graphic></inline-graphic>, −∞<x<∞<inline-graphic></inline-graphic>. In particular, we show that θˆn→Pθif and only iflimx→∞xrP(|X|>x)=∞∀r>θ.<graphic></graphic> This means that, under very reasonable conditions on F(⋅)<inline-graphic></inline-graphic>, θˆn<inline-graphic></inline-graphic> is actually a consistent estimator of <italic>θ</italic>. [ABSTRACT FROM AUTHOR]