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PutThe log-likelihood, apart from an additive constant, isOver the full parameter space its maximum-likelihood estimates are $(\hat{\mu },{\hat{\sigma }}^2)=(\overline{X},S/n)$. Under $H_0$, maximizing over the nuisance parameter $\mu$ gives $({\hat{\mu }}_0,{\sigma }_0^2)=(\overline{X},1)$. Thus the generalized likelihood-ratio test statistic, in the convention $2({\sup }_{\Theta }{\ell }_n-{\sup }_{{\Theta }_0}{\ell }_n)$, is

Under $H_0$, the normal-sample decomposition givesWriting $Z_n=(S-(n-1))/\sqrt{2(n-1)}$, the central limit theorem for a sum of squared independent standard normal variables yields $Z_n\stackrel{d}{\to }N(0,1)$. Hence Slutsky theorem givesand in particular ${\hat{\sigma }}^2\stackrel{P}{\to }1$.

For $h(x)=x-1-\log x$, Taylor theorem at $x=1$ givesConsequentlyby the continuous mapping theorem and Slutsky's theorem. Since the square of a standard normal variable has the chi-squared distribution with one degree of freedom,

In local coordinates centred at $p$ and $f(p)$, a nonconstant rational map has the formThe point $p$ is a ramification point of a holomorphic map when its ramification index of a holomorphic map satisfies $e_p>1$. A branch value of a holomorphic map, also called a branch point in the target, is a value $f(p)$ of some ramification point.

Let $B$ be the finite set of branch values and put $R=f^{-1}(B)$. For every $w\notin B$, all points of $f^{-1}(w)$ have local degree one, so the holomorphic inverse function theorem supplies disjoint neighbourhoods on which $f$ is biholomorphic. Compactness of the fibre lets their target neighbourhoods be intersected to one evenly covered neighbourhood of $w$. Henceis an unramified covering map. Here the displayed restriction requires $R=f^{-1}(B)$; if “ramification point” is reserved only for points with $e_p>1$, then the source deletion must be written $f^{-1}(B)$.

The monodromy theorem says that analytic continuations along endpoint-fixed homotopic paths have the same terminal germ. Equivalently here, the path lifting theorem lifts a loop $\gamma$ based at $w\notin B$ from each point of $f^{-1}(w)$; taking the endpoint of each lift permutes that fibre. The permutation depends only on $[\gamma ]\in {\pi }_1({\mathbb{C}}_{\infty }\setminus B,w)$, giving the monodromy group of a covering.

For the stated function, make the Möbius change of coordinateThenThe critical points are $t=0,\pm 1/\sqrt{2},\infty$, with branch valuesA loop around $0$ interchanges the two roots born from $t=0$, so its branch cycle is a transposition. A loop around $-1/4$ simultaneously interchanges the two pairs that collide there, so its branch cycle is a product of two disjoint transpositions. With a suitable labelling these areThey generate a transitive group of order eight; their product is a four-cycle. Therefore the full monodromy group isthe dihedral group in its action on the four vertices of a square.