A method for estimating the power of moments

• Published in: Journal of inequalities and applications 2018, Vol.2018 (1), p.1-14, Article 54
• Main Authors: Chang, Shuhua, Li, Deli, Qi, Yongcheng, Rosalsky, Andrew
• Format: Article
• Language: English
• Subjects: Analysis; Applications of Mathematics; Asymptotic theorems; Consistent estimator; Mathematics; Mathematics and Statistics; Point estimator; Power of moments
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-018-1645-7

Let X be an observable random variable with unknown distribution function F ( x ) = P ( X ≤ x ) , − ∞ < x < ∞ , and let θ = sup { r ≥ 0 : E | X | r < ∞ } . We call θ the power of moments of the random variable X . Let X 1 , X 2 , … , 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: θ ˆ n = log n log max 1 ≤ k ≤ n | X k | , where log x = ln ( e ∨ x ) , − ∞ < x < ∞ . In particular, we show that θ ˆ n → P θ if and only if lim x → ∞ x r P ( | X | > x ) = ∞ ∀ r > θ . This means that, under very reasonable conditions on F ( ⋅ ) , θ ˆ n is actually a consistent estimator of θ .