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Non-parametric statistics literature (E.g. Eq 2.2 in here, or DKW inequality) define empirical CDF (ECDF) by assigning PMF of $\frac{1}{n}$ to each data point. They then proceed to make use of these objects, e.g. to derive error bounds on the estimated distributions etc.

One advantage of this ECDF definition is that, unlike PDFs, one doesn't have to make explicit choice of bin width, which is known to greatly impact estimation of PDFs.

However, it is easy to see that if we chose different bin width for ECDF, our CDF estimation would differ from that obtained by assigning a PMF of $\frac{1}{n}$ to each point. This quantization effect will be more pronounced for small samples (which one could argue is not the right regime for application of non-parametric methods, but please bear with me). A different bin-width could make physical sense, for example, if our measurements were noisy or the measuring instrument is outputting more decimals than is physically meaningful.

My question is: Is the PMF of $\frac{1}{n}$ optimal for CDF estimation in any sense? How are the convergence results impacted if we were to use a wider bin-width to estimate the CDF?

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    $\begingroup$ It is optimal in that it is the discrete variable maximum likelihood estimate. That is, any other discrete distribution will lead to a lower likelihood. $\endgroup$
    – Cliff AB
    Commented May 22, 2018 at 4:47
  • $\begingroup$ @CliffAB Is it then reasonable to suspect that the MLE estimate would lead to overfitting? In a sense we're committing too much to the data locations. $\endgroup$
    – RDK
    Commented May 22, 2018 at 5:22
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    $\begingroup$ @RDK MLE is concerned about finding such estimate that fits your data best; it is not concerned about overfitting, on another hand, it gives you convergence promises: en.wikipedia.org/wiki/… $\endgroup$
    – Tim
    Commented May 22, 2018 at 8:32
  • $\begingroup$ @Tim Thank you. I meant that overfitting frequently occurs to MLE estimates as a result of relying too much on data. An oft-quoted example is estimating $P(heads)$ from 3 coin tosses which all land heads. MLE estimate would be $P(heads) = 1.0$. Would the MLE estimate of CDF be subject to a similar phenomenon in the situation that our data have noise (which they often do in the cases of interest). $\endgroup$
    – RDK
    Commented May 22, 2018 at 13:55

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