Let $Y$ be a random vector in $\mathbb{R}^k$, with distribution function belonging to a family $\{F_{\theta}, \theta\in\Theta\}$ is a parametric family of distribuitiuons (e.g. normal with unknown parameters $\theta= (\mu, \Sigma)$. You can put some simple assumptions such as continuity etc if you want).
Let $m=1$. I am looking for a function $g: \mathbb{R}^k\times \Theta \to \mathbb{R}^m$ satisfying both:
- $W:= g(Y, \theta)$ is a random variable whose distribution under $F_{\theta}$ does not depend on $\theta$. (if we put there correct $\theta$, then $W$ does not matter which $\theta$ was the correct one)
- $W_{\theta}:= g(Y, \theta)$ satisfy that $W_{\theta_1}\overset{D}{\neq}W_{\theta_2}$ for each $\theta_1\neq\theta_2\in\Theta$
For example, if $k=1$ then $g(Y, \theta):=F_{\theta}(Y)$ satisfy that $W\sim UNIF(0,1)$ (first point is ok) and $W_{\theta_1}\overset{D}{\neq}W_{\theta_2}$ if $\{F_{\theta}, \theta\in\Theta\}$ does not contain two identical functions (I think). I am trying to generalize this and find similiar result for at least $k=2$.
I have a feeling it is not possible for $m=1$. What about $m=k$? (i.e. $W\in\mathbb{R}^k$).