I have a large but finite set of objects (phylogenetic trees), each of which is assigned an integer value 0 ≤ v ≤ x. x varies from set to set, but is small (≪ 100).
For a given set, I wish to estimate p(v ≤ n) for each n in 0 → x.
p(v = 0) can be calculated exactly. Because p(v = n) is close to 1/n when n ~ x, I'm able to obtain good estimates by sampling many (a million, say) objects; the standard error can be calculated using the binomial distribution. But p(v = n) is very small when n is small: even a large sample may contain zero objects with n = 1 or n = 2.
I wish to interpolate between my precise estimates for large n and the exact value known for p(n = 0) in order to obtain estimates for p(n ≤ 1, 2, …) in a way that reflects the increasing trend in p as n increases, and to calculate the uncertainty associated with these estimates.
My approach so far has been to attempt a non-linear regression, but I've not found a way to conduct this or other interpolation techniques that I've encountered in a manner that expressly accounts for the known uncertainty of the estimated values.