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The P-star approximation describes the sampling density of a regular maximum-likelihood estimator in sample coordinates consisting of the fitted parameter $\hat{\theta }$ and a suitable ancillary statistic $a$. For a $d$-dimensional parameter, write $\ell (\theta ;\hat{\theta },a)$ for the log-likelihood expressed in these coordinates, and $j(\hat{\theta };\hat{\theta },a)$ for the fitted observed information. The approximation isThe determinant factor supplies the local volume scale, while the exponential retains the full likelihood ratio, rather than replacing it by a quadratic. The normalizing constant is chosen by integrating over the fitted-parameter coordinates; its leading regular large-sample value is $(2\pi {)}^{-d/2}$. These ingredients arise from a statistical saddlepoint approximation. Their use requires an interior, locally unique fit, nonsingular information and suitable smoothness; an ancillary statistic specifies the conditioning surface when the fit alone does not exhaust the relevant sample coordinates. In a regular model the unnormalized leading expression has relative error of order $n^{-1}$ locally; normalization and conditioning matter for refined inference. It is not a universal exact density. Under a smooth one-to-one parameter transformation, the Hessian at the fit transforms without a score term, so its determinant supplies precisely the density Jacobian determinant. This proves the invariance of the construction under one-to-one changes of statistical parameters.

Here put $S={\sum }_i{X}_i$ and $T={\sum }_i{Y}_i$. The log-likelihood, apart from a data-only constant, isThe two score equations giveThe sample sums form a sufficient statistic, and there is no need to retain an additional ancillary in calculating their joint density. Inverting the fitted coordinates gives $S=n/\hat{\lambda }$ and $T=n/(\hat{\psi }\hat{\lambda })$. With parameter order $(\psi ,\lambda )$, the fitted observed information isAlso,Thus the required joint P-star approximation isTo integrate out $\hat{\lambda }$, substitute $u=b/\hat{\lambda }$ inTaking $b=n\lambda (1+\psi /\hat{\psi })$ and rearranging the powers yieldsThe beta function identity ${\int }_0^{\infty }{z}^{n-1}(1+z{)}^{-2n}dz=\Gamma (n{)}^2/\Gamma (2n)$ therefore givesBoth normalized approximations are actually exact. To check the joint assertion independently, $S$ and $T$ have independent gamma distributions with shape $n$ and rates $\lambda$ and $\psi \lambda$. Their joint density is ${\psi }^n{\lambda }^{2n}{S}^{n-1}{T}^{n-1}{e}^{-\lambda S-\psi \lambda T}/\Gamma (n{)}^2$. The inverse-coordinate Jacobian determinant iswhich reproduces the normalized joint expression. This is the P-star density for two exponential rates; its marginal gives the stated F-distribution. Stirling's approximation gives $c_n=(2\pi {)}^{-1}(1-1/(6n)+O(n^{-2}))$, so using only the leading normalizer retains the exact shape but not its exact total mass.

The original PDF's displayed density constant is missing a square on $\Gamma (n)$. Its printed expression integrates to $\Gamma (n)$, rather than one in general. The normalized density above is the one consistent with the stated F-distribution.