Likelihood deviance for nested models on common support

Fix an observed response vector $y$ for which the component densities are positive throughout both models' parameter sets, and two stated GLMs with the same response family, base support, and log-likelihood function $\mathrm{\ell }(\beta ;y)$. They are nested here only when a stated embedding identifies the fitted parameter set with a subset ${\mathrm{\Theta }}_{\mathrm{f}\mathrm{i}\mathrm{t}}\subseteq {\mathrm{\Theta }}_{\mathrm{s}\mathrm{a}\mathrm{t}}$ of the saturated reference parameter set. Require both sets to be nonempty and both suprema to be finite real numbers. The likelihood deviance is

$D(y)=2\{{\sup }_{\beta \in {\mathrm{\Theta }}_{\mathrm{s}\mathrm{a}\mathrm{t}}}\mathrm{\ell }(\beta ;y)-{\sup }_{\beta \in {\mathrm{\Theta }}_{\mathrm{f}\mathrm{i}\mathrm{t}}}\mathrm{\ell }(\beta ;y)\}\mathrm{.}$

The definition makes no claim about deviance residuals, goodness-of-fit diagnostics, or a reference distribution. The two full-text GLM notes cited above support likelihood setup but do not supply a deviance-diagnostics treatment; the nonnegativity result below is proved only from the stated set inclusion.