Fitting distribution for count data

I am interested to fit distribution to the count data presented as histogram below:

Is there a way to understand what is the most likely distribution of the data without apriori hypothesizing the distribution?

Following, let's say I suspect the data to be distributed according to zero-inflated Poisson distribution. If I don't have any explanatory variables, is the right way to model it would be:

library(pscl)
fm_zip <- zeroinfl(span$lifespan ~1, data = counts)  EDIT: After fitting negaitve binomial zero-inflated model (lifespan ~1, ) the following model is reached (visualized with rootograms; countreg package): There seems to be a systematic overestimation in the first part of the histogram (x<20), and underestimation in the second part. • Is there a way to improve the fit without adding new explanatory variable? • Does it make sense to add dummy variable that indicates whether X>20 as explanatory variable? • What are your estimates for the mean and variance (assuming zero-inflated poisson)? Dec 31, 2016 at 12:35 • Your approach seems a good starting point. I might be a bit worried about the slight piling up at higher counts. If you get a poor fit try the negative binomial of which Poisson is a special case. The vignette for pscl is a mine of useful information. Dec 31, 2016 at 13:34 • "The most likely distribution" is the one that assigns probability$k/n$to the number$x$when$x$has appeared$k$times in a dataset of$n\$ values. In other words, it's exactly the empirical distribution of the data. This illustrates a classic trade-off: if you're willing to assume little or nothing about the data-generation process, then what you can infer from the data is equally limited.
– whuber
Dec 31, 2016 at 15:26
• @ilanman the mean is 6.31 and standard deviation is 11.39. Thanks! Jan 1, 2017 at 7:45