Let $(X_1,X_2,\cdots,X_n)$ be a random sample drawn from a population with distribution function $F$. Is the empirical distribution function $F_n$ the UMVUE of $F$? ( $F$ itself is the parameter of interest.)
For fixed $x\in\mathbb R$, the empirical DF is defined as
\begin{align} F_n(x)&=\frac{1}{n}{P(\text{number of observations}\leqslant x)} \\&=\begin{cases}0&,\text{ if }x<X_{(1)}\\\frac{r}{n}&,\text{ if }X_{(r)}\leqslant x<X_{(r+1)}\,,\quad r=1,2,\cdots,n-1\\1&,\text{ if }x\geqslant X_{(n)}\end{cases} \end{align}
We see that $nF_n(x)\sim\text{Bin}(n,F(x))$, whence
$$E(F_n(x))=F(x)$$ and $$\operatorname{Var}(F_n(x))=\frac{F(x)(1-F(x)}{n}\longrightarrow 0\quad\text{ as }n\uparrow \infty$$
Clearly, $\hat{F_n}(x)$ is an unbiased estimator of $F(x)$.
In fact, $\hat{F_n}(x)$ is a consistent estimator of $F(x)$ since for each $x\in\mathbb R$, $$\hat{F_n}(x)\stackrel{P}{\longrightarrow}F(x)\quad\text{ as }n\uparrow \infty$$
Now the question is whether $\hat F_n(x)$ itself can be seen as a complete sufficient statistic for $F(x)$. The sample $(X_1,\cdots,X_n)$ or the order statistics $(X_{(1)},\cdots,X_{(n)})$ are sufficient for $F(x)$. But what is a complete sufficient statistic for $F(x)$?
In any other problem, we can start with a trivial unbiased estimator of $F(x)$, namely $$h(X_1,\cdots,X_n)=\begin{cases}1&,\text{ if }X_1\leqslant x\\0&,\text{ otherwise }\end{cases}$$
And we find $E(h\mid T)$ where $T$ is a complete sufficient statistic for the family of distributions $\{F_{\theta}:\theta\in\Theta\}$ (i.e. $F$ parametrised by some unknown $\theta$ ). But what happens when $F$ itself is the parameter?
One motivation for this question is this old post on Math.SE:
An estimator for the c.d.f $F$ at a point $x_0$?
Maybe this is a well-known result but I could not find a proper reference. If required, it can be assumed that $F$ is continuous and a population pdf $f$ exists.
Edit.
I had not realised that this problem is in fact related to non-parametric estimation. The following is an extract from Theory of Point Estimation by Lehmann-Casella (2nd edition, pg 109).
The exercise 1.6.33 mentioned here asks to show that for i.i.d $X_i$ with unknown density $f$, the order statistics are complete.