= Solution
For an interest parameter $\psi$ and <nuisance parameter> $\lambda$, the <profile likelihood> is $L_p(\psi)=L(\psi,\widetilde\lambda_\psi)$, where $\widetilde\lambda_\psi$ maximizes the <likelihood> at fixed $\psi$. Its maximum occurs at the joint <maximum-likelihood estimate> of $\psi$. It preserves the optimized <likelihood ratios> for different interest values and is unchanged by one-to-one reparametrization of the nuisance at each fixed $\psi$. It generally is neither a marginal sampling <density> nor a conditional <likelihood>, and eliminating <nuisance parameters> by maximization can introduce appreciable small-sample distortion. Differentiating the constrained <score equation> gives the curvature formula
$$
-\ell_p''=j_{\psi\psi}-j_{\psi\lambda}j_{\lambda\lambda}^{-1}j_{\lambda\psi}
$$
at a joint stationary point. The corresponding expected-information expression is the <efficient information> for the interest parameter. Under the regular conditions of <Wilks theorem>, twice the <profile log-likelihood> drop has an asymptotic <chi-squared distribution> with the dimension of the interest parameter as its <degrees of freedom>.
For scalar nuisance, the <Barndorff-Nielsen modified profile likelihood> may be written
$$
\ell_m(\psi)=\ell_p(\psi)-\frac12\log j_{\lambda\lambda}(\psi,\widetilde\lambda_\psi)-\log\left|\frac{\partial\widetilde\lambda_\psi}{\partial\widehat\lambda}\right|,
$$
where the sample-coordinate <derivative> holds $\widehat\psi$ and the <ancillary statistic> fixed. For vector nuisance the last two quantities are <determinants>. An equivalent form is $\ell_m=\ell_p+\tfrac12\log|j_{\lambda\lambda}|-\log|\ell_{\lambda;\widehat\lambda}|$, since differentiating the constrained nuisance score gives $\ell_{\lambda;\widehat\lambda}=j_{\lambda\lambda}\partial\widetilde\lambda_\psi/\partial\widehat\lambda$. The semicolon denotes differentiation in sample coordinates, not another ordinary parameter <derivative>. This <Jacobian determinant> adjustment is essential to the definition.
In the present model set
$$
A(\psi)=\sum_iY_ie^{-\psi x_i},\quad B(\psi)=\sum_i x_iY_ie^{-\psi x_i},\quad C(\psi)=\sum_i x_i^2Y_ie^{-\psi x_i}.
$$
The <log-likelihood> is $\ell=-n\log\lambda-\psi\sum_i x_i-A(\psi)/\lambda$. Consequently
$$
\boxed{\widetilde\lambda_\psi=\frac{A(\psi)}n,\qquad\ell_p(\psi)=-n\log\frac{A(\psi)}n-\psi\sum_i x_i-n.}
$$
The profile score is $nB/A-\sum_i x_i$, and its <derivative> is $-n(C/A-(B/A)^2)$. The expression in parentheses is the <variance> of the fixed design values under positive weights proportional to $Y_ie^{-\psi x_i}$. Thus the <profile log-likelihood> is strictly concave for a nonconstant design. Its score changes from $n\max_i x_i-\sum_i x_i>0$ to $n\min_i x_i-\sum_i x_i<0$ as $\psi$ goes from $-\infty$ to $+\infty$, giving a unique finite fit. If all $x_i$ coincide, only the mean $\lambda e^{\psi x_i}$ is identifiable and inference on $\psi$ alone is not defined; that exceptional design must be excluded.
The specified ancillary coordinates give $Y_i=\widehat\lambda\exp(\widehat\psi x_i+a_i)$. At fixed $\widehat\psi,a$, therefore,
$$
\frac{\partial Y_i}{\partial\widehat\lambda}=\frac{Y_i}{\widehat\lambda},\qquad \frac{\partial\widetilde\lambda_\psi}{\partial\widehat\lambda}=\frac{\widetilde\lambda_\psi}{\widehat\lambda}.
$$
Meanwhile the constrained <observed information> is
$$
j_{\lambda\lambda}=\left[-\frac n{\lambda^2}+\frac{2A}{\lambda^3}\right]_{\lambda=\widetilde\lambda_\psi}=\frac n{\widetilde\lambda_\psi^2}.
$$
Substitution proves
$$
\boxed{\ell_m(\psi)=\ell_p(\psi)+\log\widehat\lambda-\frac12\log n=\ell_p(\psi)+\text{constant in }\psi.}
$$
Equivalently the sample <derivative> of the nuisance score is $\ell_{\lambda;\widehat\lambda}=A/(\lambda^2\widehat\lambda)$, which equals $n/(\widetilde\lambda_\psi\widehat\lambda)$ at the constrained fit and gives the same cancellation. Thus the <modified profile likelihood for exponential regression> is proportional to the ordinary <profile likelihood>. This conclusion uses the given ancillary; no proof of its ancillarity is required.
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