After some digging around, I would say there are indications that your data might follow what's called a power law, but I wouldn't sign off on it without additional inspection. The two approaches I used that lead me to suspect this particular distribution are a Pareto QQ-plot and a log-log plot.
- As written in the link, when comparing the logarithm of the sample data $X$ to the quantiles of an Exponential distribution with mean equal to 1, $Exp(1)$, then if the points "asymptotically" converge to a straight line, this indicates a power law distribution:
The catch here is that your most frequent observation is indeed $X=0$, and so the log transform of that brings its own difficulties in seeing the behavior of
your data apropos this method. If however we employ this same method in the case where we omit the zero cases from the data, then the picture becomes significantly different:
I would therefore recommend that if you are indeed binning the data into discrete buckets, at least start with $X=1$ as the minimum value.
- The other method is the log-log plot, which plots the log of the data vs the log of their observed frequencies:
The rule of thumb here is that if, for large values in the x-axis, we start observing a straight line, then this could point to a power law distribution. To reiterate the article in the link above, the catch here is that this particular method requires very large amounts of data to provide reliable results, only works with binned data (which just happens to be the case here), and of course, whether or not it starts resembling a straight line from $log(x) \approx 5.2$ and after is probably open to interpretation.
One further method you can try for identifying the underlying distribution consists of the residual quantile functions. Unfortunately, I have not worked with this type of transformation in the past to be able to provide more information. Hope this helps.