So I have this problem which I'm unsure of my answer. Any tip on how to treat it differently is more than welcome.

X and Y are independent $\mathcal{N}(\mathcal{\mu_1},\sigma^2)$ and $\mathcal{N}(\mathcal{\mu_2},\sigma^2)$, $\theta=(\mu_1,\mu_2)$ and observe $Y-X$

Is this parametrization identifiable?

I proceded as the following: Set $W= Y-X$; $\mu_0=\mu_2-\mu_1$ ; $\sigma_0^2=2\sigma^2$

I did find a pdf for $|W|$ to be : $$\mathcal{p}_{|W|}(w)= \frac{1}{\sigma_0 \sqrt{2\pi}}exp\{-\frac{w^2+\mu_0^2}{2\sigma_0^2}\}(exp(\frac{w\mu_0}{\sigma_0^2}))+(exp(\frac{w\mu_0}{\sigma_0^2}))$$

Now, setting the $\mu_0'=\mu_0-\Delta$, for real $\Delta$

and replacing $\mu_0$ in my equation would definitely yield a different pdf hence the parametrization is identifiable since with $\theta_1 \neq \theta_2 \Rightarrow P_{\theta_1} \neq P_{\theta_2}$

  • 2
    $\begingroup$ You can't tell $\mu_1=3,\mu_2=2$ from $\mu_1=5,\mu_2=4$, since they give the same $\mu_2-\mu_1$. You can identify the mean of $W$ but as I read it, the question is asking about identifiability of the parameters for $X$ and $Y$ $\endgroup$ – Glen_b Sep 29 '19 at 1:36
  • $\begingroup$ I thought they wanted to observe the difference and not each of them. You have a point since the question specify the parameter. $\endgroup$ – Mahamad A. Kanouté Sep 29 '19 at 1:48
  • $\begingroup$ From your comment, the parametrization is not identifiable then. $\endgroup$ – Mahamad A. Kanouté Sep 29 '19 at 1:49
  • $\begingroup$ If you don't mind me asking what it is really to have a parametrization that is not identifiable in the real world. I've been googling and asked questions in class but got no luck $\endgroup$ – Mahamad A. Kanouté Sep 29 '19 at 1:50
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    $\begingroup$ Not sure what "in the real world" is intended to cover, but certainly it's possible for people to try to fit models that are unidentifiable. It happens any time there's more than one value of some vector of parameters that would produce the same distribution of the data. As for googling it .... I can't imagine what you tried, but my first search turned up the relevant Wikipedia article en.wikipedia.org/wiki/Identifiability as the first hit; that in turn links to this simpler article: en.wikipedia.org/wiki/Parameter_identification_problem $\endgroup$ – Glen_b Sep 29 '19 at 1:57

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