Estimate dispersion parameters in negative binomial distribution

Question

A popular parameterization of the negative binomial distribution is by $\mu$ and $r$, which represent mean and dispersion, respectively. The probability mass function states: $$ \text{P(x = k)} = \frac{\Gamma(r + k)}{k! \Gamma(r)} \cdot \Bigg( \frac{r}{r + \mu} \Bigg)^r \Bigg( \frac{\mu}{r + \mu} \Bigg)^k $$

A common way to infer population parameter $r, \mu$ from samples is to use maximum likelihood estimation.

 
 

There is a relationship between $\mu, \sigma^2, r$, which is $$ r = \frac{\mu^2}{\sigma^2 - \mu} $$

I was wondering if this would be a reasonable way to infer population parameters:

1. calculate sample mean $\bar x$ and sample variance $s^2$
2. infer the population dispersion as $$\hat r = \frac{\bar x^2}{s^2 - \bar x}$$

Answer 1

Yes, this is a reasonable way. It is called a "Method of Moments" (MoM) estimator, as it uses the first two sample moments to calculate the estimate. Often, for the negative binomial distribution, it's actually a pretty good estimator - but by no means always.

Having said this, though, it can easily be, especially with small samples and not much overdispersion, that $s^2 \leq \bar{x}$, in which case you have a problem. In this case, the MLE doesn't exist either. Other estimators can be found in Estimating the Negative Binomial Dispersion Parameter.