A method for estimating the power of moments

• Main Authors: Shuhua Chang, Deli Li, Yongcheng Qi, Andrew Rosalsky
• Format: Article
• Language: English
• Published: SpringerOpen 2018-03-01
• Series: Journal of Inequalities and Applications
• Subjects: Asymptotic theorems; Consistent estimator; Point estimator; Power of moments
• Online Access: http://link.springer.com/article/10.1186/s13660-018-1645-7

Abstract Let X be an observable random variable with unknown distribution function F(x)=P(X≤x) $F(x) = \mathbb{P}(X \leq x)$, −∞<x<∞ $- \infty< x < \infty$, and let θ=sup{r≥0:E|X|r<∞}. $$\theta= \sup\bigl\{ r \geq0: \mathbb{E} \vert X \vert ^{r} < \infty\bigr\} . $$ We call θ the power of moments of the random variable X. Let X1,X2,…,Xn $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample of size n drawn from F(⋅) $F(\cdot)$. In this paper we propose the following simple point estimator of θ and investigate its asymptotic properties: θˆn=lognlogmax1≤k≤n|Xk|, $$\hat{\theta}_{n} = \frac{\log n}{\log\max_{1 \leq k \leq n} \vert X_{k} \vert }, $$ where logx=ln(e∨x) $\log x = \ln(e \vee x)$, −∞<x<∞ $- \infty< x < \infty$. In particular, we show that θˆn→Pθif and only iflimx→∞xrP(|X|>x)=∞∀r>θ. $$\hat{\theta}_{n} \rightarrow_{\mathbb{P}} \theta\quad\mbox{if and only if}\quad\lim_{x \rightarrow\infty} x^{r} \mathbb{P}\bigl( \vert X \vert > x\bigr) = \infty\quad\forall r > \theta. $$ This means that, under very reasonable conditions on F(⋅) $F(\cdot)$, θˆn $\hat {\theta}_{n}$ is actually a consistent estimator of θ.