= Gamma-ratio marginal and profile likelihood identity
{c}
{title2=$\ell(\psi;t)=m\log\psi-(m+n)\log(1+\psi t)$}
For independent $X\sim\operatorname{Gamma}(m,\psi\lambda)$ and $Y\sim\operatorname{Gamma}(n,\lambda)$ in shape-rate notation, $T=X/Y$ has density proportional to $\psi^m t^{m-1}(1+\psi t)^{-(m+n)}$. Its distribution is nuisance-free in lambda. The nuisance maximum is $(m+n)/(\psi X+Y)$, so substituting into the full likelihood gives the same interest-parameter log-likelihood up to a parameter-free term. Equality here is special to the common-scale structure, not a general equality of profiling and marginalization.
Back to article page
