Normal natural-mean parameter ratio estimator (${\hat{\varphi }}_1=\bar{X}/\hat{v}$)

For independent $N(\mu ,v)$ observations with unknown mean and variance, the first natural parameter is ${\varphi }_1=\mu /v$. Its maximum-likelihood estimator is $\bar{X}/[n^{-1}{\sum }_i(X_i-\bar{X}{)}^2]$. The delta method and independent asymptotic mean/variance errors give limiting variance $v^{-1}+2{\mu }^2{v}^{-2}$ for $\sqrt{n}({\hat{\varphi }}_1-{\varphi }_1)$. Knowing $v$ removes the second contribution and makes the estimator exactly normal with variance $1/(nv)$.