Profile likelihood ratios with nuisance parameters

Let $(S,\mathcal{S})$ be a measurable data space, let $\nu$ be a measure on it, and let $\mathrm{\Psi }$ and $H$ be nonempty parameter sets. For each $(\psi ,\eta )\in \mathrm{\Psi }\times H$, let $p_{\psi ,\eta }:S\to [0,+\mathrm{\infty }]$ be a specified measurable density with ${\int }_S{p}_{\psi ,\eta }d\nu =1$. A specified pointwise version of each density is part of the model because an observed-data likelihood can depend on the representative chosen within an almost-everywhere equivalence class. The coordinate $\psi$ is the parameter of interest; $\eta$ is a nuisance coordinate, meaning that it is retained in the probability model but is not the parameter whose value is tested or reported. The common dominating measure $\nu$ and the full density family are part of the model.

For observed data $s\in S$, define the likelihood $L_s(\psi ,\eta )=p_{\psi ,\eta }(s)$. Take both suprema below in the extended real line and suppose the global supremum $M_s:={\sup }_{({\psi }^{\mathrm{\prime }},{\eta }^{\mathrm{\prime }})\in \mathrm{\Psi }\times H}{L}_s({\psi }^{\mathrm{\prime }},{\eta }^{\mathrm{\prime }})$ satisfies $0<M_s<+\mathrm{\infty }$. For each $\psi \in \mathrm{\Psi }$, define the profile likelihood $M_s(\psi )={\sup }_{\eta \in H}{L}_s(\psi ,\eta )$ and the profile likelihood ratio ${\lambda }_s(\psi ):=\frac{M_s(\psi )}{M_s}\in [0,1]\mathrm{.}$ The associated likelihood-ratio statistic is $q_s(\psi ):=\{\begin{array}{ll}-2\log {\lambda }_s(\psi ),& {\lambda }_s(\psi )>0,\\ +\mathrm{\infty },& {\lambda }_s(\psi )=0,\end{array}$ where $\log$ is the real natural logarithm on $(0,\mathrm{\infty })$. These definitions use suprema and do not require a maximizing parameter value to exist. The distribution of $q_s$, a p-value calibration, and the coverage of a confidence set require additional model and sampling assumptions; none follows from this definition.

Dominated parametric likelihoods and their common dominating measure

Extended-real order and arithmetic conventions

Existence and uniqueness of extended-real suprema

Cowan et al., full likelihood-based tests paper, §2

Wasserman, full statistics textbook, Chapter 10 §10.6