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Say we are making maximum likelihood estimation of parameter $\theta$ observed data $y_1,\dots,y_n$ which are iid, a known density $f_X(\cdot; \theta)$ where $X_i = g(Y_i)$ where $g$ is a bijektiv monotone increasing or decreasing function. In this case, we can apply the change of variable to get

$$ f_Y (y;\theta) = f_X(g(y);\theta)\vert g'(y)\vert $$

which we can use to find the MLE of $\theta$. However, is this still a valid procedure if $g$ also depends on $\theta$ such that we are maximizing

$$ f_Y (y;\theta) = f_X(g(y;\theta);\theta)\vert g'(y;\theta)\vert $$

where $g(\cdot;\theta)$ is a monotone increasing or decreasing function for all values $\theta$?

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Yes, this is valid by the definition of change of variable.

For example, suppose that $y_i \sim N(\mu,1)$ and $x_i = g(y_i;\mu) = y_i-\mu \sim N(0,1)$, and therefore you cannot estimate this parameter. The problem is that you are transforming the variable based on a transformation that you do not know.

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  • $\begingroup$ Though this just happen because we substract exactly mu but would work if it was 1/2 mu that we substract then we can estimate mu. Are there a formal restriction on f and g that state whether or not we will be able to find an MLE? $\endgroup$ Dec 3, 2017 at 8:33

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