2 added 200 characters in body edited Nov 14 '17 at 17:39 Xi'an 62.9k88 gold badges101101 silver badges387387 bronze badges There is an infinite range of examples for this phenomenon since the maximum likelihood estimator of a bijective transform $$\Psi(\theta)$$ of a parameter $$\theta$$ is the bijective transform of the maximum likelihood estimator of $$\theta$$, $$\Psi(\hat{\theta}_\text{MLE})$$; the expectation of the bijective transform of the maximum likelihood estimator of $$\theta$$, $$\Psi(\hat{\theta}_\text{MLE})$$, $$\mathbb{E}[\Psi(\hat{\theta}_\text{MLE})]$$ is not the bijective transform of the expectation of the maximum likelihood estimator, $$\Psi(\mathbb{E}[\hat{\theta}_\text{MLE}])$$; most transforms $$\Psi(\theta)$$ are expectations of some transform of the data, $$\mathfrak{h}(X)$$, at least for exponential families, provided an inverse Laplace transform can be applied to them. There is an infinite range of examples for this phenomenon since the maximum likelihood estimator of a bijective transform $$\Psi(\theta)$$ of a parameter $$\theta$$ is the bijective transform of the maximum likelihood estimator of $$\theta$$, $$\Psi(\hat{\theta}_\text{MLE})$$; the expectation of the bijective transform of the maximum likelihood estimator of $$\theta$$, $$\Psi(\hat{\theta}_\text{MLE})$$, $$\mathbb{E}[\Psi(\hat{\theta}_\text{MLE})]$$ is not the bijective transform of the expectation of the maximum likelihood estimator, $$\Psi(\mathbb{E}[\hat{\theta}_\text{MLE}])$$; There is an infinite range of examples for this phenomenon since the maximum likelihood estimator of a bijective transform $$\Psi(\theta)$$ of a parameter $$\theta$$ is the bijective transform of the maximum likelihood estimator of $$\theta$$, $$\Psi(\hat{\theta}_\text{MLE})$$; the expectation of the bijective transform of the maximum likelihood estimator of $$\theta$$, $$\Psi(\hat{\theta}_\text{MLE})$$, $$\mathbb{E}[\Psi(\hat{\theta}_\text{MLE})]$$ is not the bijective transform of the expectation of the maximum likelihood estimator, $$\Psi(\mathbb{E}[\hat{\theta}_\text{MLE}])$$; most transforms $$\Psi(\theta)$$ are expectations of some transform of the data, $$\mathfrak{h}(X)$$, at least for exponential families, provided an inverse Laplace transform can be applied to them. 1 answered Nov 13 '17 at 7:13 Xi'an 62.9k88 gold badges101101 silver badges387387 bronze badges There is an infinite range of examples for this phenomenon since the maximum likelihood estimator of a bijective transform $$\Psi(\theta)$$ of a parameter $$\theta$$ is the bijective transform of the maximum likelihood estimator of $$\theta$$, $$\Psi(\hat{\theta}_\text{MLE})$$; the expectation of the bijective transform of the maximum likelihood estimator of $$\theta$$, $$\Psi(\hat{\theta}_\text{MLE})$$, $$\mathbb{E}[\Psi(\hat{\theta}_\text{MLE})]$$ is not the bijective transform of the expectation of the maximum likelihood estimator, $$\Psi(\mathbb{E}[\hat{\theta}_\text{MLE}])$$;