  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.
