$X\sim\mathcal N(a,b^2/c^2)$ can we estimate $a$, $b$, and $c$ separately?

Question

Suppose we have $X\sim\mathcal N(a,b^2/c^2)$. Using only observations of $X$, can we estimate $a$, $b$, and $c$ separately?

I believe the answer is no. Here is my logic. First note that there is an infinite number of solutions that can satisfy the distributional assumption on $X$. Specifically, saw we knew the value of $Var(X)=\sigma^2$. Notice that any combination of $b$ and $c$ of the form $b=\sigma c$ would work.

Furthermore, suppose we set up a system of three equations using the moments of $X$ as $$ E(X)=a\\ E(X^2)=a^2+b^2/c^2\\ E(X^3)=a^3+3ab^2/c^2. $$ While we have three equations and three unknowns, $b$ and $c$ always appear as a ratio and thus we cannot invert the system to solve for $a$, $b$, and $c$ separately.

Therefore, there is no way we could hope to estimate all three parameters just from observations of $X$. The best we could do is estimate $a$ and the ratio $b/c$. Am I correct? Can someone prove me right or wrong and show me a more rigorous argument?

Answer 1

You are right. We cannot estimate $b$ and $c$ separately, we can only estimate the ratio $b/c$. Different values of $b$ and $c$ lead to exactly the same distribution and hence the samples will contain, intuitively speaking, the same information even though the values of $b$ and $c$ are different.

This is the concept of identifiability which is defined in the following way in Statistical Inference by Berger and Casella (page 523, second edition, 2002).

A parameter $\theta$ for a family of distributions $\{f(x\mid\theta):\theta\in\Theta\}$ is identifiable if distinct values of $\theta$ correspond to distinct pdfs or pmfs. That is, if $\theta\ne\theta'$, then $f(x\mid\theta)$ is not the same function of $x$ as $f(x\mid\theta')$.

I hope this helps.