Is it possible that two Random Variables from the same distribution family have the same expectation and variance, but different higher moments?

Question

I was thinking about the meaning of location-scale family. My understanding is that for every $X$ member of a location scale family with parameters $a$ location and $b$ scale, then the distribution of $Z =(X-a)/b$ does not depend of any parameters and it's the same for every $X$ belonging to that family.

So my question is could you provide an example where two random from the same distribution family are standardized but that does not results in a Random Variable with the same distribution?

Say $X$ and $Y$ come from the same distribution family (where with family I mean for example both Normal or both Gamma and so on ..). Define:

$Z_1 = \dfrac{X-\mu}{\sigma}$

$Z_2 = \dfrac{Y-\mu}{\sigma}$

we know that both $Z_1$ and $Z_2$ have the same expectation and variance, $\mu_Z =0, \sigma^2_Z =1$.

But can they have different higher moments?

My attempt to answer this question is that if the distribution of $X$ and $Y$ depends on more than 2 parameters than it could be. And I am thinking about the generalized $t-student$ that has 3 parameters.

But if the number of parameters is $\le2$ and $X$ and $Y$ come from the same distribution family with the same expectation and variance, then does it mean that $Z_1$ and $Z_2$ has the same distribution (higher moments)?

Thank you

Answer 1

Let $Z\sim N(0,1)$; you can look up the formula $$\mu(a)=E|Z|^a =\frac{2^{a/2}\Gamma(\frac{a+1}2)}{\sqrt\pi},$$ for all $a>0$. Use this to construct a one parameter family $X_a$ of random variables, as follows:$$X_a=\frac{|Z|^a-\mu(a)}{\sqrt{\mu(2a)-\mu(a)^2}}.$$ Obviously the first two moments of $X_a$ are all the same, but yet their distributions all differ (because the density function of $X_a$ tapers to $0$ at $\infty$ in an $a$-dependent way).

More complicated famililies with more parameters are possible, obviously. Let $X_{a,b}$ be a recentered and rescaled version of (say) $|Z+a|^b$, and so on.

Most named parametric families of probability distributions are not location-scale families, and for any such the same trick applies: the non-central chi squared distribution, the beta distribution, the $F$ distribution. Take any of these families, recenter and rescale so the standardized versions all have the same first two moments, and what you end up with is more examples of what you are looking for.