How to make negative binomial distribution model based on data?

Question

I have a set of data that has actual frequency. I have difficulty to find the method of building the negative binomial distribution model. How to relate the parameters in negative binomial distribution with the data?

Answer 1

It is possible to reproduce the results in the table.

You can find the mean of the data is $0.03561302$ and the variance (without a Bessel correction) is $0.06156488$, so overdispersed compared to a Poisson distribution.

Since the mean of a negative binomial distribution is $\frac{(1-p)r}{p}$ and the variance is $\frac{(1-p)r}{p^2}$, the method of moments could suggest using $\hat p=0.5783977$ and $\hat r=0.04887077$. While these parameters are useful, they are not particularly meaningful in themselves: in the standard construction of the negative binomial, $r$ is the number of successes when you stop so a positive integer, but since they are just curve-fitting they might have ignored that point and that this $hat r$ is very small.

Using R to do the calculations, you then get

n <- 0:10
counts <- c(8586, 176, 35, 13, 6, 1, 0, 0, 0, 0, 0)
m <- sum(n * counts) / sum(counts)
v <- sum(n^2 * counts) / sum(counts) - m^2 
phat <- m / v
rhat <- m^2 / (v - m)
c(m=m, v=v, phat=phat, rhat=rhat) 
#          m          v       phat       rhat 
# 0.03561302 0.06156488 0.57846328 0.04887077 
cbind(n=n, 
      actual=counts, 
      negbin=round(dnbinom(n, rhat, phat) * sum(counts), 2), 
      poisson=round(dpois(n, m)* sum(counts), 2) ) 
#  n actual  negbin poisson
#  0   8586 8584.26 8508.53
#  1    176  176.84  303.01
#  2     35   39.09    5.40
#  3     13   11.25    0.06
#  4      6    3.62    0.00
#  5      1    1.23    0.00
#  6      0    0.44    0.00
#  7      0    0.16    0.00
#  8      0    0.06    0.00
#  9      0    0.02    0.00
# 10      0    0.01    0.00

which seems to match the numbers in the table