I'm interested in computing a modified profile likelihood and came across the stably adjusted profile likelihood proposed by Barndorff-Nielsen and Cox in their book 'Inference and Asymptotics' which can be found on page 270-271. However the notation is quite confusing and the examples are not so helpful in this regard.
So, suppose we have have a parameter vector that can be partitioned as $(\psi, \chi)$, where $\psi$ is the parameter of interest and $\chi$ is a vector of nuisance parameters. Let $\hat{\chi}_{\psi}$ be the value of $\chi$ that maximizes the likelihood for a fixed $\psi$ so that $\ell (\psi, \hat{\chi}_{\psi})$ is the profile log-likelihood function. Also let $i_{\chi\chi}$ and $j_{\chi \chi}$ denote the corner of the Fisher and observed information matrix respectively corresponding to the nuissance parameters. Then, the authors propose the following modification to the profile likelihood: $$ L_{A}^{*} = \left\{ \frac{|i_{\chi\chi}(\psi, \hat{\chi}_{\psi})|}{|j_{\chi\chi}(\psi, \hat{\chi}_{\psi})|} \right\}^{1/2}\exp\left\{g_{0}(\psi)\right\}L_{p}(\psi) $$
Where, $$ g_{0}(\psi) = \int_{\hat{\psi}}^{\psi}h(\psi, \hat{\chi}_{\psi}) d\psi $$ and, $$ h(\psi, \chi)=tr\left\{ i_{\chi\chi}^{-1}(\psi, \chi) H_{\chi\chi\psi} \right\} $$ $$ H_{\chi\chi\psi} = i_{\chi\psi/\chi}-i_{\chi\chi/\chi}i^{-1}_{\chi\chi}i_{\chi\psi}-\frac{1}{2}(i_{\chi \chi/\psi}-i_{\chi\chi/\chi}i^{-1}_{\chi\chi}i_{\chi\psi}) $$
Now, earlier in the book they define the notation $\theta_{/\lambda} = \frac{\partial\theta}{\partial\lambda}$. If $\chi$ is a vector and $i_{\chi\chi}$ is a matrix then presumably $i_{\chi\chi/\chi} = \frac{\partial i_{\chi\chi}}{\partial \chi}$, which is the derivative of matrix with respect to a vector, and the result would be a tensor. But then the multiplications with $i^{-1}_{\chi\chi}$ are not well-defined. So I guess my question is how to interpret the notation for $H_{\chi\chi\psi}$ and carry out the computation?
EDIT: The same formula for $H_{\chi\chi\psi}$ appears in the paper below, but again, there is no explanation of the notation.
https://www.jstor.org/stable/2346033?seq=1#metadata_info_tab_contents