If a p-th standardized absolute moment for $X$ is higher than for $Y$, is this also true for other $p$?

Question

Say the p-th standardized absolute moment of a distribution, if it exist, is:

$$\mu_{\vert p\vert}(X) = E\left( \left| \frac{X-\mu_X}{\sigma_X} \right|^p \right)$$

If for some $p>2$ we have $\mu_{\vert p\vert}(X)>\mu_{\vert p\vert}(Y)$, is this then also true for other p-th standardized absolute moments?

Answer 1

The answer is no. Distributions can be differently higher/lower in relation to each other for different moments.

Example

Consider the distribution

$$f(x,a) = \begin{cases} 0.075 & \text{if} & x = -a \\ 0.175 & \text{if} & x = -1 \\ 0.500 & \text{if} & x = 0 \\ 0.175 & \text{if} & x = 1 \\ 0.075 & \text{if} & x = a \\ \end{cases}$$

Let's plot $\mu_{\vert 4 \vert}$ (the kurtosis) and $\mu_{\vert 6 \vert}$ of $f(x,a)$ as function of $a$ and compare them to the standardized moments of a Gaussian distribution.

Here we see that for $2 \lessapprox a \lessapprox 2.3$ we have that the distribution $f(x,a)$ has a higher $\mu_{\vert 4 \vert}$ but lower $\mu_{\vert 6 \vert}$, relative to a normal distribution.