estimation - What is the proper way to estimate the CDF for a . . . The ECDF has many nice properties such as being strongly consistent (pointwise even) to the CDF Since you have a discrete approximation of a continuous distribution you can generate quantiles that can be used for confidence intervals in the usual discrete way
distributions - Empirical CDF vs CDF - Cross Validated The CDF is a theoretical construct - it is what you would see if you could take infinitely many samples The empirical CDF usually approximates the CDF quite well, especially for large samples (in fact, there are theorems about how quickly it converges to the CDF as the sample size increases)