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An exponential family has density $h(x)\exp \{\varphi \cdot T(x)-k(\varphi )\}$ on a parameter-independent support. For an independent and identically distributed sample, its likelihood factorizes through ${\sum }_{i}T(X_i)$, so the Fisher-Neyman factorization theorem gives a sufficient statistic whose dimension is the fixed dimension of $T$, independent of sample size.

Expanding the normal log-density gives ${\varphi }_1=\mu /v$, ${\varphi }_2=-1/(2v)$, andThe mean-value parameter is $\nabla k=(\mathbb{E}X,\mathbb{E}{X}^2)$, henceFor $n\ge 2$ and positive sample variance, $\hat{\mu }=\bar{X}$ and $\hat{v}=n^{-1}{\sum }_i(X_i-\bar{X}{)}^2$, so ${\hat{\Phi }}_1=\bar{X}/\hat{v}$. For this normal natural-mean parameter ratio estimator, the central limit theorem gives asymptotic independent errors for $\hat{\mu }$ and $\hat{v}$, with variances $v/n$ and $2v^2/n$. Applying the delta method to $\mu /v$ yieldsIf $v=v_0$ is known, ${\Phi }_2=-1/(2v_0)$ is fixed and ${\hat{\Phi }}_1=\bar{X}/v_0$ has exact distribution $\boxed{N({\Phi }_1,1/(n{v}_0))}$. The variance-estimation contribution $2{\mu }^2/v^2$ disappears.