These functions are primarily designed for speed in simulation. Limited
argument validation is performed.
Suppose \(X_1 \sim \text{NB}(\mu, \theta_1)\) and
\(X_2 \sim \text{NB}(r\mu, \theta_2)\) where \(X_1\) and \(X_2\) are
independent, \(X_1\) is the count outcome for items in group 1, \(X_2\)
is the count outcome for items in group 2, \(\mu\) is the arithmetic mean
count in group 1, \(r\) is the ratio of arithmetic means for group 2 with
respect to group 1, \(\theta_1\) is the dispersion parameter of group 1,
and \(\theta_2\) is the dispersion parameter of group 2.
Unequal dispersion parameters
When the dispersion parameters are not equal, the likelihood is
$$
\begin{aligned}
L(r, \mu, \theta_1, \theta_2 \mid X_1, X_2) = & \left( \frac{\theta_1^{\theta_1}}{\Gamma(\theta_1)} \right)^{n_1} \frac{\mu^{\sum{x_{1i}}}}{(\mu + \theta_1)^{\sum{x_{1i} + n_1 \theta_1}}} \times \\
& \left( \frac{\theta_2^{\theta_2}}{\Gamma(\theta_2)} \right)^{n_2} \frac{(r \mu)^{\sum{x_{2j}}}}{(r \mu + \theta_2)^{\sum{x_{2j} + n_2 \theta_2}}} \times \\
& \prod_{i = 1}^{n_1}{\frac{\Gamma(x_{1i} + \theta_1)}{x_{1i}!}} \prod_{j = 1}^{n_2}{\frac{\Gamma(x_{2j} + \theta_2)}{x_{2j}!}}
\end{aligned}
$$
and the parameter space is
\(\Theta = \left\{ (r, \mu, \theta_1, \theta_2) : r, \mu, \theta_1, \theta_2 > 0 \right\}\).
The log-likelihood is
$$
\begin{aligned}
l(r, \mu, \theta_1, \theta_2) = \ &n_1 \left[ \theta_1 \ln \theta_1 - \ln \Gamma(\theta_1) \right] + \\
&n_2 \left[ \theta_2 \ln \theta_2 - \ln \Gamma(\theta_2) \right] + \\
&(n_1 \bar{x}_1 + n_2 \bar{x}_2) \ln(\mu) - n_1 (\bar{x}_1 + \theta_1) \ln(\mu + \theta_1) + \\
&n_2 \bar{x}_2 \ln(r) - n_2 (\bar{x}_2 + \theta_2) \ln(r \mu + \theta_2) + \\
&\sum_{i = 1}^{n_1}{\left( \ln \Gamma(x_{1i} + \theta_1) - \ln(x_{1i}!) \right)} + \\
&\sum_{j = 1}^{n_2}{\left( \ln \Gamma(x_{2j} + \theta_2) - \ln(x_{2j}!) \right)}
\end{aligned}
$$
Equal dispersion parameters
When the dispersion parameters are equal, the likelihood is
$$
\begin{aligned}
L(r, \mu, \theta \mid X_1, X_2) = & \left( \frac{\theta^{\theta}}{\Gamma(\theta)} \right)^{n_1 + n_2} \times \\
& \frac{\mu^{\sum{x_{1i}}}}{(\mu + \theta)^{\sum{x_{1i} + n_1 \theta}}} \frac{(r \mu)^{\sum{x_{2j}}}}{(r \mu + \theta)^{\sum{x_{2j} + n_2 \theta}}} \times \\
& \prod_{i = 1}^{n_1}{\frac{\Gamma(x_{1i} + \theta)}{x_{1i}!}} \prod_{j = 1}^{n_2}{\frac{\Gamma(x_{2j} + \theta)}{x_{2j}!}}
\end{aligned}
$$
and the parameter space is
\(\Theta = \left\{ (r, \mu, \theta) : r, \mu, \theta > 0 \right\}\).
The log-likelihood is
$$
\begin{aligned}
l(r, \mu, \theta) = \ &(n_1 + n_2) \left[ \theta \ln \theta - \ln \Gamma(\theta) \right] + \\
&(n_1 \bar{x}_1 + n_2 \bar{x}_2) \ln(\mu) - n_1 (\bar{x}_1 + \theta) \ln(\mu + \theta) + \\
&n_2 \bar{x}_2 \ln(r) - n_2 (\bar{x}_2 + \theta) \ln(r \mu + \theta) + \\
&\sum_{i = 1}^{n_1}{\left( \ln \Gamma(x_{1i} + \theta) - \ln(x_{1i}!) \right)} + \\
&\sum_{j = 1}^{n_2}{\left( \ln \Gamma(x_{2j} + \theta) - \ln(x_{2j}!) \right)}
\end{aligned}
$$