# MLE of a function of parameter

I am trying to get my head around the maximum likelihood estimator (MLE) of a function of a parameter.

Say I have $$X_i \sim \text{Poisson}(\theta)$$ samples. I want to find the MLE of $$\pi(\theta) = \exp(-\theta) = P_\theta(X = 0)$$.

MLE of the Poisson's parameter is the sample average, i.e. $$\bar{X}$$. Is

$$\exp \left( -\bar{X} \right)$$ the MLE of $$\pi$$ ?

Let us assume that the sample $$X_1,\ldots,X_n$$ is independently and identically distributed (iid), i.e. $$X_i\overset{\text{iid}}{\sim} \text{Poisson}(\theta),\quad i=1,\ldots,n,$$

and let $$\pi = e^{-\theta}$$. Then $$\theta = -\log\pi$$ and thus the likelihood function for $$\pi$$ is

$$L(\pi) \propto e^{n\log\pi} (-\log \pi)^{\sum_i X_i}.$$

The log-likelihood function is $$\ell(\pi) = n\log\pi + \sum_iX_i\log(-\log\pi),$$

and the maximum likelihood estimator (MLE) is the solution in $$\pi$$ of

$$\ell^\prime(\pi) = 0 = \frac{n}{\pi} + \frac{\sum_i X_i}{\log\pi}\frac{1}{\pi}.$$

The MLE is thus $$\log\hat\pi = -\bar X$$ or $$\hat\pi = e^{-\bar X}$$. But this comes by no surprise since:

the MLE is invariant with respect to reparametrizations.


The claim that the MLE is invariant holds even when the reparametrization is not one-to-one, though in this case, the proof is more involved.

If $T$ is a statistic, which is MLE for parameter $\theta$, and $f$ is a continuous one-to-one function then $f(T)$ is the MLE of $f(\theta)$. Prove this by using transformation of variables formula for probability distributions and the definition of MLE. For a quick overview see here.

• f does not need to be continuous or one-to-one. It holds for any f. Jul 11, 2019 at 4:29
• @JamesYang it is a theorem for $f$ as stated but it is an axiom for other $f.$ In other words, if $f$ is not one-to-one, and $\hat \theta$ is the MLE of $\theta,$ then, by definition, $f(\hat \theta)$ is the MLE of $f(\theta).$ Nov 15, 2022 at 20:41