Negative binomial mean-size parameter orthogonality ($I_{\mu \theta }=0$)

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.