I am trying to prove that if I have two samples of iid random variables, then MLE's based on these two samples will be also independent.
More formally, let
$$\mathbf{x} = (x_i)^T_{i = 1} \stackrel{iid}{\sim} \mathcal{N}(0,1), \quad \mathbf{y} = (y_i)^T_{i = 1} \stackrel{iid}{\sim} \mathcal{N}(0,1)$$ be two samples of the same size. Let
$$\theta_1 = \text{argmax}_{\theta} \mathcal{L(\mathbf{x}, \theta)}, \quad \theta_2 = \text{argmax}_{\theta} \mathcal{L(\mathbf{y}, \theta)}$$
be two MLE estimates of the same parameter $\theta$.
Problem: I want to prove that $\theta_1$ and $\theta_2$ are independent. Intuitively it is easy to see, because there are no chances for them to be dependent since the underlying samples are generated by iid random variables. But I am struggling to show this formally.
Any support will be appreciated.
If two random variables $X$ and $Y$ are independent, then for any (measurable) maps $f$ and $g$, $g(X)$ and $g(Y)$ are independent. This is a straightforward deduction of any definition you take for independence.
$\theta_1$ is a function of $\mathbf{x}$, $\theta_2$ is a function of $\mathbf{y}$, and $\mathbf{x}$ and $\mathbf{y}$ are independent. Therefore $\theta_1$ and $\theta_2$ are independent.