Empirical conditional cumulative distribution function

Question

I have got a question on how to get conditional cumulative distribution (regarding another),

To be clearer, let us consider two random variables, X, Y, of samples xi and yi We can estimate the cdf by $\hat{F}_X(x)=\sum_{i=1}^{n}I_{x_i\leq X}$

I am looking to have an estimator of $F_{X|y}(x)$ (one variable in condition or more),

Do you know how to get emprical values of such cdf?

Many thanks in advance

Answer 1

The simplest way is to just calculate the joint ECDF $$\mathbb{F}_{X,Y}(x,y) = P(X \le x, Y \le y)=\sum_{i=1}^n\mathcal{I}(X_i < x)\mathcal{I}(Y_i < y).$$

Then $$\mathbb{F}_{X|Y=y_0}(x)=\mathbb{F}_{X,Y}(x, y=y_0)$$