The stated mean and variance identify $\lambda_i$ as a <gamma distribution> with shape $\nu_i$ and scale $\alpha_i$. A <Poisson-gamma mixture> is negative binomial, with
$$
\mathbb E Y_i=\mu_i=\alpha_i\nu_i,
\qquad
\operatorname{Var}(Y_i)=\mu_i+\frac{\mu_i^2}{\nu_i}.
$$
When $\nu_i=\nu$, this is exactly the constant-shape variance model fitted by the negative binomial regression, so the most appropriate printed p-value is $0.0798$ from model3.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 218 / 1 / e / i / Solution

{c}

If $Y\mid\Lambda\sim\operatorname{Poisson}(\Lambda)$ and $\Lambda$ has a <gamma distribution>, then $Y$ has a <negative binomial distribution>. For gamma shape $\nu$ and scale $\alpha$,
$$
\mathbb EY=\alpha\nu,
\qquad
\operatorname{Var}(Y)=\alpha\nu+\alpha^2\nu.
$$

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 Incoming links: Poisson-gamma mixture