= P-star density for an exponential mean
{title2=$p^*(v;\theta)\propto v^{n-1}\theta^{-n}e^{-nv/\theta}$}
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 $n^nv^{n-1}e^{-nv/\theta}/(\Gamma(n)\theta^n)$. This illustrates an exact transformation-model case rather than a general claim of exactness.
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