Abstract
Let X be an observable random variable with unknown distribution function F(x) = P(X≤ x) , − ∞ OpenSPiltSPi xOpenSPiltSPi ∞ , and let θ=sup{r≥0:E|X|rOpenSPiltSPi∞}.We call θ the power of moments of the random variable X. Let X1, X2, … , Xn 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: θˆn=lognlogmax1≤k≤n|Xk|where log x= ln (e∨ x) , − ∞ OpenSPiltSPi xOpenSPiltSPi ∞. In particular, we show that θˆn→Pθif and only iflimx→∞xrP(|X|CloseSPigtSPix)=∞∀rCloseSPigtSPiθ. This means that, under very reasonable conditions on F(⋅) , θˆ n is actually a consistent estimator of θ.
| Original language | English (US) |
|---|---|
| Article number | 54 |
| Journal | Journal of Inequalities and Applications |
| Volume | 2018 |
| DOIs | |
| State | Published - 2018 |
Bibliographical note
Funding Information:The authors are grateful to the referee for carefully reading the manuscript and for offering helpful suggestions and constructive criticism which enabled them to improve the paper. The research of Shuhua Chang was partially supported by the National Natural Science Foundation of China (Grant #: 91430108 and 11771322) and the research of Deli Li was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada (Grant #: RGPIN-2014-05428).
Publisher Copyright:
© 2018, The Author(s).
Keywords
- Asymptotic theorems
- Consistent estimator
- Point estimator
- Power of moments