# Why is $X$ not an identifiable statistical model

In my textbook, Identifiablity is defined as so:

For any $$\theta_1, \theta_2 \in \Theta$$ , if $$\theta_1 \neq \theta_2 \Rightarrow \Bbb P_{\theta_1} \neq \Bbb P_{\theta_2}$$ , where $$\Bbb P_{\theta}$$ is the probability distribution function.

Then it says that $$X \sim N(\frac{\theta_1}{\theta_2}, \theta_3)$$ where $$\theta_1 \in \Bbb R$$ , $$\theta_2, \theta_3 \gt 0$$ does not define an identifiable statistical model. Why?

• Consider, for example, $(\theta_1,\theta_2,\theta_3)=(0,1,1)$ and $(\theta_1,\theta_2,\theta_3)=(0,2,1)$. Both triples correspond to the same $\text{N}(0,1)$ distribution. – Zen Jan 11 '20 at 16:58
• How am I supposed to use the definition to prove this example? Since in the definition there are only 2 $\theta$'s. This is my main concern – The Poor Jew Jan 11 '20 at 17:04
• Do we suppose that $\theta_1' = (\theta_1, \theta_2, \theta_3)$ and then if $\theta_1' = (0,1,1) \neq \theta_1''=(0,2,1)$ does not imply inequality of $\Bbb P_{\theta_1'}$ and $\Bbb P_{\theta_1"}$ ? Hence model is not identifiable? – The Poor Jew Jan 11 '20 at 17:15
• Yes, also related is this question on identification – Jesper for President Jan 11 '20 at 17:59
• Please add the self-study tag and read its wiki. – kjetil b halvorsen Jan 12 '20 at 14:43