I would like to estimate the distribution of a very large population of known size but unknown mean and variance. I cannot assume anything about the shape of the underlying distribution (although I am relatively certain that it is not normal). However, I am certain that the values in the population are non-negative, non-zero integers. I believe that the distribution is naturally lumped in some way discretely rather than being continuously distributed over the entire range. I cannot sample the entire population but I would like to estimate the probability density function. I would also like some level of assurance of the correctness of the estimated distribution. What is the best way to go about this?
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Since the empirical distribution function of the sample should converge to the true population distribution function as the sample size approaches the population size, it seems to me that best estimate of the population pdf is the empirical pdf of the largest sample you an afford to draw.
Depending upon what the sample pdf looks like, you could consider summarizing the distribution with some functional form, particularly if you have some ideas about what generated the population.