Example of a maximum likelihood estimator that is not a sufficient statistic

Question

I am currently researching on providing some bounds on estimation using some information theoretic tools (I won't expend on that here for now, I may make a post about it later) and turns out that given a phenomenon $X$, an observation $Y$, then $\hat{x}(Y)$, the maximum likelihood estimator of $X$ based on $Y$, may apparently not be a sufficient statistic and this is a something I would like to study, the answer to this post states that such an example exists when $Y$ consists of samples that are not i.i.d but fails to provide such an example and I haven't found anything about it. Have anyone seen something of the sort ?

Answer 1

If $T$ is a sufficient statistic for $\theta$ and a unique MLE of $\theta$ exists, then the MLE must be a function of $T$.

So if you can find a situation where there can be several maximum likelihood estimators, there remains a possibility that you can choose one MLE that might not be a function of a sufficient statistic alone.

A simple example to consider is the $U(\theta,\theta+1)$ distribution.

Consider i.i.d random variables $X_1,X_2,\ldots,X_n$ having the above distribution.

Then the likelihood function given the sample $(x_1,\ldots,x_n)$ is

$$L(\theta)=\prod_{i=1}^n \mathbf1_{\theta<x_i<\theta+1}=\mathbf1_{\theta<x_{(1)},x_{(n)}<\theta+1}\quad,\,\theta\in\mathbb R$$

A sufficient statistic for $\theta$ is $$T(X_1,\ldots,X_n)=(X_{(1)},X_{(n)})$$

And an MLE of $\theta$ is any $\hat\theta$ satisfying $$\hat\theta<X_{(1)},X_{(n)}<\hat\theta+1$$

or equivalently, $$X_{(n)}-1<\hat\theta<X_{(1)} \tag{1}$$

Choose $$\hat\theta'= (\sin^2 X_1)(X_{(n)}-1) + (\cos^2 X_1)(X_{(1)})$$

Then $\hat\theta'$ satisfies $(1)$ but it does not depend on the sample only through $T$.