P-star density for two exponential rates ($p^{\ast }(\hat{\psi },\hat{\lambda })$)

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 $(2n,2n)$.