Let $F_X$ be a CDF of an unknown random variable $X$. If we have independent samples $x_1, x_2, \ldots, x_n$ of $X$ then we can estimate $F_X$ non-parametrically using an ECDF $\hat{F}_n$. By Central Limit Theorem, for each $i \in \{1,2,\ldots,n\}$
$$\sqrt{n}(\hat{F_n}(x_i) - F_X(x_i)) \rightarrow N(0,F_X(x_i)[1-F_X(x_i)])$$
What I want to know : How to pick one estimated CDF using the distribution above?
Aside from $\hat{F}_n$, I'm also interested in other estimated CDFs based on the distribution per $x$. Independent sampling for each $x$ will most likely violate the non-decreasing requirement of a CDF and exceed the range $[0,1]$.
If we do this parametrically, I just need the likelihood of each parameter given the samples then use a sampling method to pick a specific set parameters to get the CDF. However, I don't have any clue what the distribution is so I'm using a non-parametric approach.