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I have cumulative counts with respect to a variable x, which looks like:

 x  cum_count
 0        100
 5        103
10        127
...       ...
Inf       187

From this, I want to arrive at an estimated pdf, maybe using kernel density estimation. How can it be done? Note that the Inf is actually present in the data.

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    $\begingroup$ Is x continuous or discrete? Are the x-values in your data supposed to represent ranges (in the continuous case)? If the values are discrete, you need a pmf rather than a pdf. Also, if the values are discrete, will we always get integer multiples of 5 only? $\endgroup$
    – Carl
    Apr 21, 2020 at 7:36
  • $\begingroup$ Are the choices for the values of the points on the cumulative CDF exactly the same as the observed values? E.g.can we deduce that there are 3 values $x = 5$. Or are the points arbitrary, unrelated to the observed values, and should we deduce that there are 3 values with $0 < x \leq 5$? $\endgroup$
    – Sextus Empiricus
    Dec 15, 2022 at 7:11

1 Answer 1

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In my opinion you can:
1) Convert your data into:

 x  empirical CDF 
 0        100/187
 5        103/187
10        127/187
...       ...
Inf       187/187

2) Using a parametric method: that is you assume a form (a distribution) for you variable and you then use maximum likelihood to estimate your parameters

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  • $\begingroup$ The likelihood function is based on a comparison with the probability density function. How do you use this with points on an emperical CDF? $\endgroup$
    – Sextus Empiricus
    Dec 15, 2022 at 7:16

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