= Solution
Partition the <statistical parameter> as $\theta=(\psi,\lambda)$, with $\psi$ of interest. For each fixed $\psi$, maximize over the <nuisance parameter> to obtain $\widehat\lambda_\psi$. The <profile likelihood> and <profile log-likelihood> are
$$
L_p(\psi)=L(\psi,\widehat\lambda_\psi),\qquad
\ell_p(\psi)=\ell(\psi,\widehat\lambda_\psi).
$$
Their maximizer is the interest component of the joint <maximum-likelihood estimator>. At a regular nuisance maximum, $\ell_p'(\psi)=\ell_\psi(\psi,\widehat\lambda_\psi)$ and its negative <Hessian matrix> is the <Schur complement>
$$
j_p=j_{\psi\psi}-j_{\psi\lambda}j_{\lambda\lambda}^{-1}j_{\lambda\psi}.
$$
The profile <likelihood ratio> is unchanged by a one-to-one nuisance reparametrization, and transforms naturally under a one-to-one interest reparametrization. Under the regular conditions of <Wilks theorem>, $2\{\ell_p(\widehat\psi)-\ell_p(\psi_0)\}$ has a limiting <chi-squared distribution> with dimension of $\psi$ degrees of freedom. Profiling is maximization, rather than integration or conditioning; $L_p$ need not be a normalized data density, and nuisance fitting can cause appreciable higher-order distortion.
The <modified profile likelihood> corrects for nuisance curvature and sample-coordinate distortion. In suitable coordinates $(\widehat\psi,\widehat\lambda,a)$, with $a$ an <ancillary statistic>, set
$$
J_\psi=\left|\det\frac{\partial\widehat\lambda_\psi}{\partial\widehat\lambda}\right|,
$$
holding $\widehat\psi,a$ fixed when differentiating the data coordinates. Up to an interest-independent factor, the <Barndorff-Nielsen modified profile likelihood> is
$$
\boxed{L_m(\psi)=L_p(\psi)
|j_{\lambda\lambda}(\psi,\widehat\lambda_\psi)|^{-1/2}J_\psi^{-1}.}
$$
Equivalently, implicit differentiation of the nuisance score equation gives $J_\psi=|j_{\lambda\lambda}|^{-1}|\ell_{\lambda;\widehat\lambda}|$, and
$$
\ell_m=\ell_p+\tfrac12\log|j_{\lambda\lambda}|
-\log|\ell_{\lambda;\widehat\lambda}|.
$$
The semicolon denotes sample-coordinate differentiation, not a second parameter derivative. In orthogonal coordinates, the <Cox-Reid adjusted profile likelihood> uses the information adjustment alone; it agrees with this modification when $J_\psi=1$, as happens below.
For the <Inverse Gaussian distribution>, write $s=\sum_i y_i$, $r=\sum_i y_i^{-1}$ and $\bar y=s/n$. Terms independent of both parameters can be absorbed into $c(y)$, giving
$$
\ell(\psi,\lambda)=c(y)+\frac n2\log\psi
-\frac\psi2\left(r-\frac{2n}{\lambda}+\frac{s}{\lambda^2}\right).
$$
The nuisance <score function> is
$$
\ell_\lambda=\frac{\psi(s-n\lambda)}{\lambda^3}.
$$
It is positive below $\bar y$ and negative above it. Thus its global maximum is
$$
\widehat\lambda_\psi=\widehat\lambda=\bar y,
$$
independent of $\psi$. Put $D=r-n/\bar y$. Then
$$
\boxed{\ell_p(\psi)=\frac n2\log\psi-\frac\psi2D+\text{constant}.}
$$
The nuisance <observed information> at this fit is
$$
j_{\lambda\lambda}(\psi,\bar y)=\frac{n\psi}{\bar y^3}.
$$
Since the constrained nuisance estimator is literally the full nuisance estimator as a function of the data, $J_\psi=1$ in the specified sample coordinates. Hence the <modified profile likelihood for inverse Gaussian shape> is
$$
\boxed{\ell_m(\psi)=\frac{n-1}{2}\log\psi-\frac\psi2D+\text{constant}.}
$$
For $n\ge2$ and a nonconstant positive sample, the <Cauchy-Schwarz inequality> gives $D>0$. The corresponding fits are $\widehat\psi_p=n/D$ and $\widehat\psi_m=(n-1)/D$. A constant sample has $D=0$ and no finite profile maximum; under continuous independent sampling this event has probability zero when $n\ge2$. In particular, the modified coefficient accounts for the fitted mean <nuisance parameter>.
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