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A method for estimating the power of moments

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... Full description

Journal Title: Journal of Inequalities and Applications 2018, Vol.2018(1), pp.1-14
Main Author: Chang, Shuhua
Other Authors: Li, Deli , Qi, Yongcheng , Rosalsky, Andrew
Format: Electronic Article Electronic Article
Language: English
Subjects:
ID: E-ISSN: 1029-242X ; DOI: 10.1186/s13660-018-1645-7
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recordid: springer_jour10.1186/s13660-018-1645-7
title: A method for estimating the power of moments
format: Article
creator:
  • Chang, Shuhua
  • Li, Deli
  • Qi, Yongcheng
  • Rosalsky, Andrew
subjects:
  • Asymptotic theorems
  • Consistent estimator
  • Point estimator
  • Power of moments
ispartof: Journal of Inequalities and Applications, 2018, Vol.2018(1), pp.1-14
description: 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 X 1 , X 2 , … , X n $X_{1}, X_{2}, \ldots, X_{n}$ 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: θ ˆ n = log n log max 1 ≤ k ≤ n | X k | , $$\hat{\theta}_{n} = \frac{\log n}{\log\max_{1 \leq k \leq n} \vert X_{k} \vert }, $$ where log x = ln ( e ∨ x ) $\log x = \ln(e \vee x)$ , − ∞ < x < ∞ $- \infty< x < \infty$ . In particular, we show that θ ˆ n → P θ if and only if lim x → ∞ x r P ( | 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 ( ⋅ ) $F(\cdot)$ , θ ˆ n $\hat {\theta}_{n}$ is actually a consistent estimator of θ .
language: eng
source:
identifier: E-ISSN: 1029-242X ; DOI: 10.1186/s13660-018-1645-7
fulltext: fulltext_linktorsrc
issn:
  • 1029-242X
  • 1029242X
url: Link


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