I'd like to fit some standard distributions, say lognormal, to a set of data which unfortunately has ties. The precision of the measurement simply isn't high enough. Of course you can still fit the data but all goodness-of-fit test like KS and the like throw warnings - which is natural, since this case cannot happen for continuous distributions.
I tried to add some arbitrary small random $\epsilon$ to every data point to remove the ties without altering the data too much. As expected this results in ridiculously low p-values.
On the other hand: The data consists of some measurements per year. Taking only the average value/year one can fit a distribution very well. So I think it's really because of the ties.
Is there any magic solution to this problem?