P-star approximation ($p^{\ast }(\hat{\theta }\mid a;\theta )=c(\theta ,a)|j(\hat{\theta }){|}^{1/2}{e}^{\ell (\theta )-\ell (\hat{\theta })}$)

For a regular model with a suitable ancillary statistic $a$, the displayed expression approximates the conditional density of the maximum-likelihood estimator. The normalizing factor includes $(2\pi {)}^{-p/2}$ for a $p$-dimensional estimator, depending on convention. The observed information is evaluated at the sample's own fitted value. In a pure scale model the factor is proportional to $1/\hat{\theta }$; this Jacobian determinant can make the normalized approximation coincide with the exact conditional density.

For independent samples of size $n$ from exponential distributions with rates $\lambda$ and $\psi \lambda$, the maximum-likelihood estimators are $\hat{\lambda }=n/S$ and $\hat{\psi }=S/T$, where $S$ and $T$ are the respective sample sums. Their normalized P-star approximation is exact:Indeed, $S$ and $T$ have independent gamma distributions with shapes $n$ and respective rates $\lambda ,\psi \lambda$. The Jacobian determinant of their inverse transformation is $n^2/({\hat{\psi }}^2{\hat{\lambda }}^3)$, giving the displayed density. Alternatively the square root of the determinant of the fitted observed information is $n/(\hat{\psi }\hat{\lambda })$; multiplying by the exponential of the fitted log-likelihood difference gives the same shape. Its normalizing constant in the P-star approximation is $c_n=n^{2n-1}{e}^{-2n}/\Gamma (n{)}^2$. Integrating over $\hat{\lambda }$ givesso $\hat{\psi }/\psi$ has the F-distribution with degrees of freedom $(2n,2n)$.

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.

Fix an ancillary statistic $a$ and a parameter value at which the score function is evaluated. Express that score as a function $u(v,a)$ of the varying maximum-likelihood estimator coordinate $v$. The ordinary density change-of-variables formula applied to the P-star approximation gives the displayed density; several monotone branches require a sum. Its derivative is a sample-coordinate derivative, not generally the parameter observed information.