Let $X$ be an observable random variable with unknown distribution function $F(x) = \mathbb{ P}(X \leq x), - \infty < x < \infty$, and let \[\ \theta = \sup\left \{ r \geq 0:~ \mathbb{ E}|X|^{ r} < \infty \right \}. \] 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: \[ \hat{ \theta}_{ n} = \frac{ \log n}{ \log \max_{ 1 \leq k \leq n} |X_{ k}|}, \] where $\log x = \ln(e \vee x), ~- \infty < x < \infty$. In particular, we show that \[ \hat{ \theta}_{ n} \rightarrow_{ \mathbb{ P}} \theta~~\mbox{ if and only if}~~ \lim_{ x \rightarrow \infty} x^{ r} \mathbb{ P}(|X| > x) = \infty ~~\forall~r > \theta. \] 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.
Categories: math.PR