Is being "free" of a variable and independent of it the same thing?

Question

I am dealing with a question to do with Maximum Likelihood estimates for a 2-parameter exponential distribution, (sample of size n, $x_i$~ Exp($\theta$, $\eta$)).

I am required to show $\hat \theta$ and $\hat \eta$, the respective MLEs, are independent. I have got values of $\hat \eta$ = $x_{1:n}$ and $\hat \theta$ = $\bar x - x_{1:n}$.

I have also found the distribution of $\bar x- \eta $ to be free of $\eta$, and I'm wondering is this enough to say that the MLEs are independent?

Answer 1

You should check whether $$ f_{X_{(1)}}(x) f_{\hat{\theta}} (y) = f_{ X_{(1)}, \hat\theta}(x,y). $$ Try to show that $$ f_{ X_{(1)}, \hat\theta}(x,y) = f_{\hat{\theta}}(y|x)f_{X_{(1)}}(x) = f_{\hat{\theta}}(y) f_{X_{(1)}}(x), $$ if I remember it right , it should be the "cleanest" way.