# Fit cdf and pdf from points on empirical cdf

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.

• 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?
– Carl
Commented Apr 21, 2020 at 7:36
• 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 $0 < x \leq 5$? Commented Dec 15, 2022 at 7:11

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

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