# Picking a specific estimated CDF from a set of CDFs provided by an ECDF

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.

$$\hat{F}_n(x) \equiv \frac{1}{n} \sum_{i=1}^n \mathbb{I}(x_i \leqslant x).$$
It is simple to establish that this function satisfies the requirements of a distribution function for all $$\mathbf{x}_n \in \mathbb{R}^n$$, so you are incorrect to think that any of these properties will be violated merely because the sample values are IID given the underlying distribution function $$F$$.
Now, you are correct that the CLT gives the asymptotic normal distribution you have stated. A small variation in this statement allows you to get a pivotal quantity for the estimation problem, which will allow you to form a confidence interval for $$F(x)$$ using the estimated value $$\hat{F}_n(x)$$. This is essentially just a confidence interval for a proportion using binomial data, so standard forms apply. (There are a number of different forms for confidence intervals for binomial proportions, but I am a fan of the Wilson score interval.) That is what we use this distributional result for --- it does not imply that we would use a different set of data for the estimate than the data that is actually observed.