# Estimation of CDF in multiple points

Suppose we have a sample $$X_1, \ldots, X_n$$ of i.i.d. real-valued random variables with an (unknown cumulative) distribution $$F$$.

The goal is to estimate the value of $$F$$ in multiple points. That is, given a set of $$k$$ numbers $$x_1 < x_2 < \ldots < x_k$$, we want to derive a set of pairs $$\{ (l_i, u_i) \}_{i \in [1,k]}$$ such that $$\mathbb{P}[\forall i \in [1,k] : l_i \le F(x_i) \le u_i] \ge 1 - \alpha,$$ where $$1 - \alpha$$ is the required coverage probability.

If $$k = 1$$, then one can use a binomial proportion confidence interval (to determine the proportion of elements that are smaller than $$x_k$$). If $$k$$ is large, then one can use Dvoretzky–Kiefer–Wolfowitz inequality (DKW).

However, what if $$k$$ is reasonably small? Let's say that $$k \in [2, 5]$$.

• I know that in this case DKW inequality may be needlessly wide. So, probably, it is not the best approach.
• At the same time, we cannot use (at least in theory) a bunch of binomial proportion confidence intervals (one interval for each $$x_i$$), since the estimates would be correlated and thus $$\prod_{i \in [1,k]} \mathbb{P}[l_i \le F(x_i) \le u_i] \not= \mathbb{P}[\forall i \in [1,k] : l_i \le F(x_i) \le u_i].$$
• I was thinking about applying simultaneous confidence intervals for multinomial proportions. However, note that they give estimates for proportion of elements falling into $$[x_i, x_{i+1})$$, whereas I need an estimation of proportion of elements falling into $$(-\infty, x_{i})$$ for each $$i$$.

1. What is the right approach here? Are there any papers on this topic?
2. Does the problem changes if $$X_1, \ldots, X_n$$ is a sample of i.i.d. integer-valued random variables?
• You seem to be asking about a generalization of the Kolmogorov-Smirnov statistic. Are you familiar with the theory of that test? – whuber Sep 28 '20 at 23:29
• No, I'm not familiar with this test. Is your idea to view the input sample as interval-censored data and then to apply some method to derive CDF? – Sergey Bozhko Sep 30 '20 at 21:33