Difference between odd central moments and even central moments
When we compute $n$-th central moments
$$\mu_n = \int_{-\infty}^\infty (x-\mu)^nf(x) dx$$
then there is a difference in interpretation of the cases when $n$ is odd or even.
For even $n$ the expression $(x-\mu)^n$ is always positive.
This central moment describes the spread of the values in the distribution away from the moment, without considering whether the value $(x-\mu)$ is negative or not.
For odd $n$ the expression $(x-\mu)^n$ is positive for $x>\mu$ and negative for $x<\mu$.
The odd moments incorporate some difference between the left and right side of the distribution (and the even moments do not).
Making odd central moments like even central moments.
There is a definition of moments where the odd moments get the same type of interpretation as the even moments, and that is when we have the absolute central moment (which occurs for instance here).
$$\mu^\prime_n = \int_{-\infty}^\infty |x-\mu|^nf(x) dx$$
for even $n$ this expression is the same as the regular central moment $\mu^\prime_n = \mu_n$.
Making even central moments like odd central moments.
We could also turn it the other way around and make even central moments with the negative and positive components in a similar way as the odd central moments, by including the sign in the expression.
$$\mu^\dagger_n = \int_{-\infty}^\infty \text{sign}(x-\mu) |x-\mu|^nf(x) dx$$
for odd $n$ this expression is the same as the regular central moment $\mu^\dagger_n = \mu_n$.
Some applications might be when $n=0$ in which case $\mu^\dagger_n$ expresses the difference in the probability mass left and right of the mean. I imagine that there might have been uses for $n=2$ or higher as well. For instance as an alternative for skewness.
Has this type of pseudo-moment been used before and are there names for it?
