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