I have four true false question about the relation between MLE(maximum likelihood estimator) and minimal sufficient statistic. let a minimal sufficient statistic exists and denoted by $S(x)$. let $x_1, \cdots, x_n$ is a random sample from a distribution $f_\theta(\cdot )$.I use notation $x=(x_1, \cdots, x_n).$
Q1) The maximum likelihood estimator always is a function of minimal sufficient statistic. true or false?
Q2) "MLE always is a function of the minimal sufficient statistic, if MLE is unique". true or false?
Q3) If MLE is not unique, so we can find an MLE such that is not a function of the minimal sufficient statistic. true or false?
Q4) If MLE is not unique, so we can find an MLE such that is a function of the minimal sufficient statistic. true or false?
Since $S(x)$ is a minimal sufficient statistic so We can write $f(x_1, \cdots, x_n,\theta)=h(x)g(S(x),\theta)$ and we expected MLE be is not a function of $S(x)$.
I think an example could help us to answer the questions. Let $x_1, \cdots, x_n$ is a random sample from $Uniform(\theta -\frac{1}{2}, \theta +\frac{1}{2})$.
So a minimal sufficient statistic is $S=(x_{(1)}=\min (x_1, \cdots, x_n),x_{(n)}=\max (x_1, \cdots, x_n)).$
$$L(\theta)=f_\theta(x_1,\cdots , x_n)=1_{\left( x_{(n)}-\frac{1}{2},x_{(1)}+\frac{1}{2}\right)}(\theta).$$
So any value in $\left( x_{(n)}-\frac{1}{2},x_{(1)}+\frac{1}{2}\right)$ is an MLE of $\theta$, like $\frac{x_{(1)}+x_{(n)}}{2}$.
Let define $W=\cos^2(x_2) (x_{(n)}-\frac{1}{2})+\sin^2(x_2) (x_{(1)}+\frac{1}{2})$. So it is an MLE too, but is not a function of the minimal sufficient statistic.
Q1:"The maximum likelihood estimator always is a function of sufficient statistic" is not true. $W=\cos^2(x_2) (x_{(n)}-\frac{1}{2})+\sin^2(x_2) (x_{(1)}+\frac{1}{2})$ is a counter example
Q2)"If MLE is unique, MLE is always a function of the minimal sufficient statistic" is true.
I am not sure here but Since $S(x)$ is a minimal sufficient statistic so We can write $f(x,\theta)=h(x)g(S(x),\theta)$ and we expected MLE be a function of $S(x)$, since maximization of $f(x,\theta)$ depend on $g(S(x),\theta)$ and $g$ depend on $x$ only through $S(x)$ .
Q3): "If MLE is not unique, so we can find an MLE such that is not a function of the minimal sufficient statistic" is true. $W=\cos^2(x_2) (x_{(n)}-\frac{1}{2})+\sin^2(x_2) (x_{(1)}+\frac{1}{2})$ is an example.
Q4) "If MLE is not unique, so we can find an MLE such that is a function of the minimal sufficient statistic" is true. $\frac{x_{(1)}+x_{(n)}}{2}$ is an example.