= Negative binomial mean-size parameter orthogonality
{title2=$I_{\mu\theta}=0$}
For the <negative binomial distribution> with <mean> $\mu>0$ and size $\theta>0$, the one-observation <score function> in the mean is $u_\mu=\theta(y-\mu)/[\mu(\mu+\theta)]$. Its size derivative is $(y-\mu)/(\mu+\theta)^2$, whose <expectation> is zero. Thus the expected <Fisher information matrix> is diagonal in $(\mu,\theta)$. Regular interior <maximum-likelihood estimators> have zero asymptotic cross covariance, without necessarily being independent in finite samples.
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