= Solution
A <marginal statistical likelihood> uses a statistic whose marginal sampling distribution depends on the parameter of interest but does not involve the <nuisance parameter>. It integrates out other observed coordinates. A <profile likelihood> instead keeps the full data and, for every fixed interest parameter, fits the <nuisance parameter> by maximizing the full <likelihood>. Profiling supplies a <likelihood> function but usually not a normalized sampling <probability density function>. Here the marginalization is of data coordinates; Bayesian model evidence averages parameters against a prior and is a separate construction.
Use shape-rate notation for the <gamma distribution>. The independent sums have $X\sim\operatorname{Gamma}(m,\psi\lambda)$ and $Y\sim\operatorname{Gamma}(n,\lambda)$, so their <joint probability density> for $x,y>0$ is
$$
f_{X,Y}(x,y)=\frac{\psi^m\lambda^{m+n}}{\Gamma(m)\Gamma(n)}x^{m-1}y^{n-1}e^{-\lambda(\psi x+y)}.
$$
For $x=tu$, $y=u$ the absolute <Jacobian determinant> is u. Therefore
$$
f_{T,U}(t,u)=\frac{\psi^m\lambda^{m+n}}{\Gamma(m)\Gamma(n)}t^{m-1}u^{m+n-1}e^{-\lambda(1+\psi t)u},\qquad t,u>0.
$$
Integrate u using the gamma integral $\int_0^\infty u^{m+n-1}e^{-bu}du=\Gamma(m+n)b^{-(m+n)}$. The lambda factors cancel, leaving
$$
\boxed{f_T(t)=\frac{\Gamma(m+n)}{\Gamma(m)\Gamma(n)}\frac{\psi^m t^{m-1}}{(1+\psi t)^{m+n}},\qquad t>0.}
$$
Thus T gives a nuisance-free marginal <likelihood>. Discarding only terms independent of psi, its logarithm is
$$
\boxed{\ell_m(\psi;t)=m\log\psi-(m+n)\log(1+\psi t).}
$$
The factor t in the logarithm is required, as in the original PDF.
The full <log-likelihood> is $m\log\psi+(m+n)\log\lambda-\lambda(\psi x+y)+c(x,y)$. Its lambda derivative vanishes uniquely at
$$
\boxed{\widehat\lambda_\psi=\frac{m+n}{\psi x+y}.}
$$
The second derivative in lambda is negative. Substituting this maximum gives $\ell_p(\psi)=m\log\psi-(m+n)\log(\psi x+y)+c'(x,y)$. Since $\psi x+y=y(1+\psi t)$, it agrees with $\ell_m(\psi;t)$ up to a psi-independent constant. This proves the <Gamma-ratio marginal and profile likelihood identity>. Their common maximizer is $\widehat\psi=m/(nt)=my/(nx)$, if an interest-parameter estimate is also wanted.
Back to article page
