P-star density for an exponential mean ($p^{\ast }(v;\theta )\propto v^{n-1}{\theta }^{-n}{e}^{-nv/\theta }$)

For a size-$n$ sample from an exponential distribution of mean $\theta$, the likelihood in the maximum-likelihood estimator coordinate $v=\bar{Y}$ has difference $\ell (\theta ;v)-\ell (v;v)=n\log (v/\theta )-nv/\theta +n$, and the fitted observed information is $n/v^2$. Their P-star approximation is the displayed gamma density shape. Its normalization gives exactly the sampling density $n^n{v}^{n-1}{e}^{-nv/\theta }/(\Gamma (n){\theta }^n)$. This illustrates an exact transformation-model case rather than a general claim of exactness.