Finding a distribution for data in $\mathbb{N}_0$

Question

Suppose, we have a set of 10,000 individuals. Each individual falls into exactly one of 200 categories. [Edit: The categories are phenotypes (different potential outcomes) of the one property that is observed]. Observing the individuals, we can count for each category the number of corresponding individuals. This number is always a non-negative integer. Suppose, that around 30 categories are empty.

Now I would like the find a distribution describing the number of individuals in each category. I did some tests with a lognormal distribution using R and the qqplots seemed to perform well.

However, I did not know how to respect the zero values which are clearly forbidden for lognormal distributed data. During my tests, I modelled the 'zero' phenomenon separately.

To this purpose, I cut off the zero values and then fitted the lognormal distribution. Going reverse (i.e. generating numbers according to the original distribution), I first used random numbers from the binomial distribution with $n=1$, deciding whether I get a zero or a non-zero. Second, I used numbers from the fitted lognormal distribution for the non-zero values.

1. Is the lognormal approach generally a bad idea for data with zeroes?
2. How would you evaluate my approach?
3. Are there better ways to find a distribution for discrete, non-negative data?

Answer 1

Your data is the classic use case for the multinomial distribution.

The binomial distribution models $N$ independent and identical 'experiments' with outcomes in $\{0, 1\}$, where "1" is often taken to indicate "success," and probability of success $p$ that is the same in all experiments.

The multinomial distribution is its generalization to $K$ integer outcomes, $\{0, \dots, k\}$, and outcome probabilities $\{p_1, \dots, p_K\}$ and $p_0 = 1 - \sum_{k=1}^K p_k$. Your data follows a multinomial distribution with $N = 10,000$, $K = 199$ (taking one category arbitrarily as a baseline). You would like to estimate the vector of probabilities $p$.

The most intuitive estimator for each probability is just the fraction of the sample that falls into each category. If $N_k$ is the number of individuals observed to be in category $k$, then $\widehat{p_k} = \frac{N_k}{N}$. This also happens to be the maximum likelihood estimator.

Think of rolling a 200-sided die 10,000 times.