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Say I have n samples $x_1,...,x_n \sim F$ where $F$ is a continuous distribution on $\mathbb R$. The Empirical Distribution Function gives us an estimation for the CDF of $F$. I am trying to understand how the corresponding pdf will look. I see in the question Empirical PDF from Empirical CDF that someone answered, that the PDF in this case will simply put a probability mass of $1/N$ at each data point $x_i$.

But if we are in the continuous case, we cannot simply define that the pdf at each point is $1/N$ and zero elsewhere, since this will give probability 0 to everything.

The pdf is the derivative of the CDF, but in our case the CDF is a step function and therefore doesn't have a derivative in the interesting points (only has a derivative of 0). So how does the PDF look in this case?

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  • $\begingroup$ I presume you ask for the PDF of the empirical distribution, rather than the PDF of the empirical PDF which, as a (step) function over $\mathbb R$, does not have a density. $\endgroup$
    – Xi'an
    Commented Apr 28 at 13:38
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    $\begingroup$ There is no standard PDF estimate that follows from the empirical CDF, since it cannot be differentiated in a useful manner, being a step function. $\endgroup$
    – Xi'an
    Commented Apr 28 at 13:41
  • $\begingroup$ @Xi'an Thanks that makes sense. $\endgroup$
    – BinyaminR
    Commented Apr 28 at 14:07

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