MLE is not invariant under parameter transformations because it's just the argmax of the likelihood!
Take for example x~normal and exp(x)~lognormal. The maximum of the distribution is at mu for the former and at exp(mu-sigma^2) for the latter, instead of exp(mu).
Take for example x~normal and exp(x)~lognormal. The maximum of the distribution is at mu for the former and at exp(mu-sigma^2) for the latter, instead of exp(mu).