Estimate parameters of an unknown negative binomial distribution based on known distribution The PDF of a known NBD given in Equation (1). The parameter a and r are function of $μ$ = sample mean, and $s^2$ = sample variance, as given in Equation (2) and (3) respectively. $r$ = number of failures, $p$ = probability of success
$P(Y=y)=((r+y-1)¦y) a^r b^y               ;   (y=0, 1, 2,……..)----(1)$
$a=μ/s^2  ----(2)$
$r=μ^2/(s^2-μ) ----(3)$
If we have an unknown NBD distribution with known mean (let’s say, $μ'$) but all other parameters are unknown (like variance $(s')^2$ is unknown). How do I estimate $a'$ and $r'$ for this unknown distribution?
Question
Can we simply take the ratio of $a/ a'$ and $r/ r'$ assuming variance ($s^2$  or $(s')^2$) is a function of mean ($μ$ or $μ'$) and the function parameter can be estimated through regression?

You can re-parameterise the negative-binomial distribution in terms of its mean $\mu$ and variance $\sigma^2$.  To do this, you use the parameter relationships:
$$p = \frac{\sigma^2 - \mu}{\sigma^2}
\quad \quad \quad 
r = \frac{\mu^2}{\sigma^2 - \mu}.$$
The re-parameterisation gives the density function:
$$\begin{align}
\text{NegBin}(y|\mu, \sigma^2)
&= \text{NegBin} \bigg( y \bigg| p = \tfrac{\sigma^2 - \mu}{\sigma^2}, r = \tfrac{\mu^2}{\sigma^2 - \mu} \bigg) \\[6pt]
&= \frac{\Gamma ( y+r )}{y! \Gamma ( r )} ( 1-p )^y p^r \\[6pt]
&= \frac{\Gamma \Big( y+\tfrac{\mu^2}{\sigma^2 - \mu} \Big)}{y! \Gamma \Big( \tfrac{\mu^2}{\sigma^2 - \mu} \Big)} \bigg( \tfrac{\mu}{\sigma^2} \bigg)^y \bigg( \tfrac{\sigma^2 - \mu}{\sigma^2} \bigg)^{\tfrac{\mu^2}{\sigma^2 - \mu}}. \\[6pt]
\end{align}$$
In the case where the mean is known, this gives a distribution parameterised by the single parameter $\sigma^2 \geqslant \mu$.
