Ways to calculate the moments of a function of sample moments?

1. Are there ways to analytically derive the moments of a function of sample moments? For example, my recent question here hasn't been addressed satisfactorily yet here: link
2. Do the moments of a function of sample moments depend on the sample distribution only through the moments of the sample distribution? This is the case in the linked question.
• The answer to your second question is in the negative. Consider, as a counterexample, the location-scale family of Cauchy distributions and let the function be (say) the ratio of the square of the second moment to the fourth moment. This ratio is always between $0$ and $1$ and therefore has a moment, but none of the distributions in this family has any moment of order $1, 2, 3,\ldots$ at all.
– whuber
Commented May 16, 2022 at 17:13

Every random variable has what is called a "moment generating function" that is an equivalent expression of the variable's probability distribution. For example, a Bernoulli random variable can be expressed by its moment generating function, where the $t^{th}$ moment equals $M_{x}(t)=1-p+pe^{t}$.
Once you find the moment generating function for the random variable of interest (for common distributions you can just look this up), if you want to learn about the moments of a function of that variable you can just transform the moment generating function into a compound function. That compound function $f(M_{x}(t))$, will give you the moments of the random variable of interest.
This is the analytical approach. Sampling matters insofar as you need to estimate the sufficient statistics for the distribution of the random variable (eg $p$ in the bernoulli case, or the mean and variance in the case of a normal distribution) in order to calculate a sample moment from a moment generating function
• So for example, if I want to learn about the moments of the sample kurtosis when sampling from a $t_{12}$ distribution (which is an example the kind of thing I think the OP is asking about), I won't be able to use an MGF for that. Commented Sep 9, 2014 at 21:01