= Modified profile likelihood for exponential regression
{title2=$\ell_m(\psi)$}
For independent <exponential distributions> with means $\lambda e^{\psi x_i}$ and a nonconstant fixed design, put $A(\psi)=\sum_iY_ie^{-\psi x_i}$. The constrained <maximum-likelihood estimate> is $\widetilde\lambda_\psi=A(\psi)/n$ and the <profile log-likelihood> is $\ell_p(\psi)=-n\log[A(\psi)/n]-\psi\sum_i x_i-n$. Use the <ancillary statistic> $a_i=\log Y_i-\log\widehat\lambda-\widehat\psi x_i$ to specify sample coordinates. At fixed $\widehat\psi,a$, the <Jacobian determinant> $\partial\widetilde\lambda_\psi/\partial\widehat\lambda=\widetilde\lambda_\psi/\widehat\lambda$, while $j_{\lambda\lambda}(\psi,\widetilde\lambda_\psi)=n/\widetilde\lambda_\psi^2$. Therefore the <modified profile likelihood> satisfies
$$
\ell_m(\psi)=\ell_p(\psi)-\tfrac12\log j_{\lambda\lambda}-\log\left|\frac{\partial\widetilde\lambda_\psi}{\partial\widehat\lambda}\right|=\ell_p(\psi)+\log\widehat\lambda-\tfrac12\log n.
$$
The adjustment is independent of $\psi$. Keeping only the <observed information> adjustment would miss this cancellation.
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