There is a known relationship stating that finding MLE is asymptotically the same is minimizing Kullback–Leibler divergence (see wiki here), or just the cross entropy. I'm wondering if there is a similar relationship in terms of a finite sample statement. Specifically, I'm wondering if it can be shown that MLE is the same as finding theta hat that minimizes the probability P with it, with the empirical CDF (wiki) using the data. That is, if we'll check the kl divergence between the ecdf of the data, and the estimated cdf of each observation based on the estimated parameter theta, then minimizing this distance will give us the MLE. I have a feeling that this can be proven using the alternative formulation of the expectancy in terms of the CDF (see wiki).
I couldn't find this in the literature from my searches, I'd be happy if someone knows how to prove this and can share (or share a link to a reference). Thanks!