A method for estimating the power of moments

作者：Chang, Shuhua; Li, Deli; Qi, Yongcheng; Rosalsky, Andrew^*

来源：JOURNAL OF INEQUALITIES AND APPLICATIONS, 2018, 2018(1): 54.

DOI：10.1186/s13660-018-1645-7

• Asymptotic theorems
• Consistent estimator
• Point estimator
• Power of moments

Let X be an observable random variable with unknown distribution function F(x) =P(X <= x), -infinity < x < infinity, and let theta = sup{r >= 0 :E broken vertical bar X broken vertical bar(r) < infinity}.
We call theta 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 theta and investigate its asymptotic properties:
theta(n) = log n /log max(1 <= k <= n) broken vertical bar X-k broken vertical bar'
I where log x = In (e v x), infinity < x < infinity. In particular, we show that
theta(n)-> p(theta) if and only if lim x'P(IXI > = infinity Vr > 0. x,co This means that, under very reasonable conditions on F(.), on is actually a consistent estimator of theta.

• 出版日期2018-3-6
• 单位天津财经大学