Rate function of an exponential sample mean ($I(x)=\lambda x-1-\log (\lambda x)(x>0)$)

For independent and identically distributed exponential random variables of rate $\lambda >0$, the sample mean has large-deviation speed $n$ and the displayed good rate function, infinite for $x\le 0$. The cumulant-generating function is $\log [\lambda /(\lambda -\theta )]$ for $\theta <\lambda$. Its Legendre-Fenchel transform has maximizing parameter $\theta =\lambda -1/x$. For $x\le 0$, sending $\theta \to -\infty$ gives infinite cost. The cost has its unique zero at $1/\lambda$ and diverges at both ends of its positive domain.