Moment aliasing in a shared zero-inflated count model ($\kappa =(\tau +\pi )/(1-\pi )$)

In a shared zero-inflated Gamma-Poisson count model, write $m_j=(1-\pi ){\mu }_j$ and $\kappa =(\tau +\pi )/(1-\pi )$. Its first two moments are $\mathrm{Var}(Y_j)=m_j+\kappa m_j^2$ and $\mathrm{Cov}(Y_j,Y_k)=\kappa m_j{m}_k$. These moments do not separately identify $\pi$, $\tau$, and the component-mean intercept: taking ${\pi }^{\prime }=0$, ${\tau }^{\prime }=\kappa$ and ${\mu }_j^{\prime }=m_j$ gives the same first two moments. The full count distribution can carry information absent from the moments. A consistent estimator of a marginal mean ratio therefore need not consistently estimate a structural-zero fraction from a misspecified moment parameterization.