A Simple Point Estimator of the Power of Moments

Abstract

Let $X$ be an observable random variable with unknown distribution function $F(x)=\mathbb{P}(X\le x),-\infty <x<\infty$, and let \[\ θ= \sup\left \{ r \geq 0:~ \mathbb{E}|X|^{r} < \infty \right \}. \] 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 $F(\cdot )$. In this paper we propose the following simple point estimator of $\theta$ and investigate its asymptotic properties: \[ \hatθ_{n} = \frac{\log n}{\log \max_{1 \leq k \leq n} |X_{k}|}, \] where $\log x=\ln (e\vee x),-\infty <x<\infty$. In particular, we show that \[ \hatθ_{n} \rightarrow_{\mathbb{P}} θ~~\mbox{if and only if}~~ \lim_{x \rightarrow \infty} x^{r} \mathbb{P}(|X| > x) = \infty ~~\forall~r > θ. \] This means that, under very reasonable conditions on $F(\cdot )$, ${\hat{\theta }}_n$ is actually a consistent estimator of $\theta$. Hypothesis testing for the power of moments is conducted and, as an application of our main results, the formula for finding the p-value of the test is given. In addition, a theoretical application of our main results is provided together with three illustrative examples.

22 pages