I have a random variable $Z$ ~ $N(\mu,\sigma)$ for which I can compute a likelihood using the normal pdf with mean = mu and standard deviation = sigma.
I could also standardize $Z$ into $Z^*$ by doing $Z^* = (Z-\mu)/\sigma$, for which I can then also compute a likelihood using the normal pdf with mean = 0 and std = 1.
The results are not the same. That is, $normpdf(Z,\mu,\sigma)$ is not equal to $normpdf(Z^*,0,1)$. My understanding is that this is normal, but please confirm.
My question is: would a Maxmimum Likelihood Estimation of the parameters yield the same results for both even though the (log) likelihood is not the same?
Ignoring the MLE part of the question, the change in density function for a location-scale change is not difficult, though with a normal distribution you have to be careful not to confuse the variance and standard deviation
$\hat Z$ is the maximum likelihood estimator in the original case if and only if $\hat Z^* =(\hat Z-\mu)/\sigma $ is the maximum likelihood estimator in the standardised case. This property is called functional equivalence and works for any bijective function