= Moment aliasing in a shared zero-inflated count model
{title2=$\kappa=(\tau+\pi)/(1-\pi)$}
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 $\operatorname{Var}(Y_j)=m_j+\kappa m_j^2$ and $\operatorname{Cov}(Y_j,Y_k)=\kappa m_jm_k$. These moments do not separately identify $\pi$, $\tau$, and the component-mean intercept: taking $\pi'=0$, $\tau'=\kappa$ and $\mu'_j=m_j$ 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.
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