Goodness of fit for exponential distribution and large sample I'm new to statistics and statistics are not my area of research so maybe my question is simple and answer is on the surface. In my research the empirical CDF for the data looks like exponential distribution CDF. Plus the values of the mean and the standard deviation are close. The data is time periods in days, so it would make sense that it is exponentially distributed.
I've tried to perform goodness of fit tests such as Chi-square test and KS test, but the problem is that the sample is large (2000+) and those tests have rejected the distribution type hypothesis. I've read that they seem to work "too" well on large samples and reject even the smallest deviation. They seem to work correctly on most smaller random samples of the main sample (200+).
What would be the best scientifically acceptable approach in this situation to:

*

*Proof that the exponential distribution can be used for the data;

*Find the most fitting distribution parameter (lambda) with 95% probability.

EDIT: I need to clarify what my research is, I should've stated that in the first place.
Without going into details of my research, the values in the dataset, let's call them T, represent harm to a person's health (measured in days spent on sick leave due to influence of production factors).
My goal is to find a way to measure probability of harm to health based on T (the bigger the T is, the higher the probability). In other words, I need to find a function of T and the result of that function would be the probability.
The idea is to use the CDF of the data as that function.
On the graph below blue line is the ECDF, red is exponential CDF.


You're basically asking "does the deviations from the fitted exponential distribution matter?" and the obvious follow-up question to that is "...matter for what?"
I can't tell you whether the deviations matter because I don't know how you will use the result!
For example, if the deviations lie in the tail of the distribution and your application is primarily influenced by the head, then no, the deviations don't matter. If your application is heavily dependent on the tail, then yeah, deviations are a problem.
I would try using both the fitted exponential distribution and your real data in the same way, and see if the difference in final outcome matters clinically for your application.
In the same way, you find the appropriate lambda by trying many different ones and seeing which one generates the smallest error in the real application compared to the real data set.

Personally, I'm biased toward using the observed, real data set as it is, rather than first fitting a theoretical distribution to it. Using the real data won't let you derive nice, clean theoretical results, but for the most part, you can easily simulate the same thing instead. And by drawing from the real data set, the results are definitely properly influenced by whatever hidden quirks you never thought to model with the theoretical distributions. Sometimes those quirks are where the money is.
