Normal likelihood with variance equal to squared mean ($\hat{\mu }=(\sqrt{S_1^2+4n{S}_2}-S_1)/(2n)$)

For independent normal distribution observations with mean $\mu >0$ and variance ${\mu }^2$, put $S_1={\sum }_i{X}_i$, $S_2={\sum }_i{X}_i^2$. The Fisher-Neyman factorization theorem makes $(S_1,S_2)$ a sufficient statistic, since the likelihood is proportional to ${\mu }^{-n}\exp [-S_2/(2{\mu }^2)+S_1/\mu ]$. Its logarithmic derivative has numerator $S_2-S_1\mu -n{\mu }^2$. When $S_2>0$, this changes sign once on the positive half-line and gives the displayed global maximum-likelihood estimator. The zero sample is a probability-zero exception with unbounded likelihood as $\mu \downarrow 0$.