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For independent exponential distributions with means $\lambda e^{\psi x_i}$ and a nonconstant fixed design, put $A(\psi )={\sum }_i{Y}_i{e}^{-\psi x_i}$. The constrained maximum-likelihood estimate is ${\tilde{\lambda }}_{\psi }=A(\psi )/n$ and the profile log-likelihood is ${\ell }_p(\psi )=-n\log [A(\psi )/n]-\psi {\sum }_i{x}_i-n$. Use the ancillary statistic $a_i=\log Y_i-\log \hat{\lambda }-\hat{\psi }{x}_i$ to specify sample coordinates. At fixed $\hat{\psi },a$, the Jacobian determinant $\partial {\tilde{\lambda }}_{\psi }/\partial \hat{\lambda }={\tilde{\lambda }}_{\psi }/\hat{\lambda }$, while $j_{\lambda \lambda }(\psi ,{\tilde{\lambda }}_{\psi })=n/{\tilde{\lambda }}_{\psi }^2$. Therefore the modified profile likelihood satisfiesThe adjustment is independent of $\psi$. Keeping only the observed information adjustment would miss this cancellation.

For independent Inverse Gaussian distributions with shape $\psi$ and mean $\lambda$, the constrained and full nuisance maximum-likelihood estimates both equal $\bar{Y}$, giving a unit sample-coordinate Jacobian determinant. The nuisance observed information there is $n\psi /{\bar{Y}}^3$. Hence subtracting half its logarithm from the profile log-likelihood replaces the coefficient $n/2$ of $\log \psi$ by $(n-1)/2$. For a nonconstant sample the modified fit is $(n-1)/({\sum }_i{Y}_i^{-1}-n/\bar{Y})$.

Under the diagonal covariance restriction, the profile log-likelihood separates into $p$ terms,Differentiating each term gives the restricted maximum-likelihood estimator ${\hat{V}}_0=D=\mathrm{diag}(S_{11},\dots ,S_{pp})$. Both fitted trace terms equal $p$, so the likelihood-ratio test statistic iswhere $R=D^{-1/2}S{D}^{-1/2}$ is the sample correlation matrix. The statistic is nonnegative by the Hadamard inequality; large values indicate dependence. Equivalently, the likelihood ratio is $(\det S/\det D{)}^{n/2}$ and rejection is for small values of this ratio.

For fixed $p$ and a positive-definite true covariance, the Wilks theorem says that twice the maximized log-likelihood difference for nested regular models converges under the null to a chi-squared distribution with degrees of freedom equal to the dimension difference. The unrestricted covariance has $p(p+1)/2$ parameters and the restricted covariance has $p$; the unknown mean has the same $p$ parameters in both. Therefore the Gaussian covariance diagonality likelihood-ratio test of asymptotic level $\alpha$ isWithin the multivariate normal distribution, diagonal covariance also means independent coordinates.

Partition the statistical parameter as $\theta =(\psi ,\lambda )$, with $\psi$ of interest. For each fixed $\psi$, maximize over the nuisance parameter to obtain ${\hat{\lambda }}_{\psi }$. The profile likelihood and profile log-likelihood areTheir maximizer is the interest component of the joint maximum-likelihood estimator. At a regular nuisance maximum, ${\ell }_p^{\prime }(\psi )={\ell }_{\psi }(\psi ,{\hat{\lambda }}_{\psi })$ and its negative Hessian matrix is the Schur complementThe 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, $2\{{\ell }_p(\hat{\psi })-{\ell }_p({\psi }_0)\}$ 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 $(\hat{\psi },\hat{\lambda },a)$, with $a$ an ancillary statistic, setholding $\hat{\psi },a$ fixed when differentiating the data coordinates. Up to an interest-independent factor, the Barndorff-Nielsen modified profile likelihood isEquivalently, implicit differentiation of the nuisance score equation gives $J_{\psi }=|j_{\lambda \lambda }{|}^{-1}|{\ell }_{\lambda ;\hat{\lambda }}|$, andThe 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 $c(y)$, givingThe nuisance score function isIt is positive below $\bar{y}$ and negative above it. Thus its global maximum isindependent of $\psi$. Put $D=r-n/\bar{y}$. ThenThe nuisance observed information at this fit isSince 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 isFor $n\ge 2$ and a nonconstant positive sample, the Cauchy-Schwarz inequality gives $D>0$. The corresponding fits are ${\hat{\psi }}_p=n/D$ and ${\hat{\psi }}_m=(n-1)/D$. A constant sample has $D=0$ and no finite profile maximum; under continuous independent sampling this event has probability zero when $n\ge 2$. In particular, the modified coefficient accounts for the fitted mean nuisance parameter.

For an interest parameter $\psi$ and nuisance parameter $\lambda$, the profile likelihood is $L_p(\psi )=L(\psi ,{\tilde{\lambda }}_{\psi })$, where ${\tilde{\lambda }}_{\psi }$ 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 formulaat 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 writtenwhere the sample-coordinate derivative holds $\hat{\psi }$ and the ancillary statistic fixed. For vector nuisance the last two quantities are determinants. An equivalent form is ${\ell }_m={\ell }_p+\frac{1}{2}\log |j_{\lambda \lambda }|-\log |{\ell }_{\lambda ;\hat{\lambda }}|$, since differentiating the constrained nuisance score gives ${\ell }_{\lambda ;\hat{\lambda }}=j_{\lambda \lambda }\partial {\tilde{\lambda }}_{\psi }/\partial \hat{\lambda }$. 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 setThe log-likelihood is $\ell =-n\log \lambda -\psi {\sum }_i{x}_i-A(\psi )/\lambda$. ConsequentlyThe 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 $Y_i{e}^{-\psi x_i}$. 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 $Y_i=\hat{\lambda }\exp (\hat{\psi }{x}_i+a_i)$. At fixed $\hat{\psi },a$, therefore,Meanwhile the constrained observed information isSubstitution provesEquivalently the sample derivative of the nuisance score is ${\ell }_{\lambda ;\hat{\lambda }}=A/({\lambda }^2\hat{\lambda })$, which equals $n/({\tilde{\lambda }}_{\psi }\hat{\lambda })$ 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.

Partition a regular parameter as $(\psi ,\lambda )$, with interest $\psi$ and nuisance $\lambda$. Orthogonal statistical parameters have zero cross-block of expected Fisher information,This may hold at one parameter value or throughout a parametrization. It is expected-information orthogonality, and does not generally force the observed information cross-block to vanish in each sample.

First, the asymptotic covariance of the joint maximum-likelihood estimator is the inverse information. Orthogonality makes this matrix block diagonal, giving first-order asymptotic independence of interest and nuisance fits. The efficient information with unknown nuisance isFor orthogonal parameters it equals $I_{\psi \psi }$. Thus estimating the nuisance creates no first-order information loss relative to knowing it, at the orthogonal point. This is a local asymptotic statement, not a general finite-sample independence theorem.

Second, orthogonality stabilizes constrained nuisance estimates. Differentiating ${\ell }_{\lambda }(\psi ,{\tilde{\lambda }}_{\psi })=0$ givesWhen the expected cross-information vanishes, regular sampling fluctuations give $j_{\lambda \psi }=O_p(\sqrt{n})$ and $j_{\lambda \lambda }=O_p(n)$. The derivative is therefore $O_p(n^{-1/2})$ locally. Changing the interest parameter by $O_p(n^{-1/2})$ changes its constrained nuisance fit by only $O_p(n^{-1})$, smaller than the ordinary nuisance estimation error. This reduces sensitivity to how the nuisance is fitted and simplifies higher-order likelihood adjustments.

For scalar interest, local orthogonal nuisance reparametrization can often be constructed explicitly. Keep $\psi$ and write the original nuisance as $\lambda (\psi ,\eta )$. The transformed interest score is $U_{\psi }+{\lambda }_{\psi }^{\mathsf{T}}{U}_{\lambda }$. Its covariance with the new nuisance score vanishes whenWith smooth nonsingular nuisance information, this ordinary differential equation has a local solution for suitable initial nuisance coordinates. For several interest coordinates the corresponding equations need compatibility conditions; global orthogonality is not automatic.

The normal model gives a transparent example. For independent observations with mean $\mu$ and variance $\nu$, direct score expectations giveThe cross-information is zero because the centered normal third moment is zero. When fitting $\nu$ at fixed $\mu$,so the change is indeed quadratic in a local change of the mean.

Orthogonality is also useful for removing nuisance effects beyond first order. Take variance $\nu$ as interest and mean $\mu$ as nuisance. Put $Q={\sum }_i(X_i-\overline{X}{)}^2$. The profile log-likelihood is ${\ell }_p(\nu )=-\frac{n}{2}\log \nu -Q/(2\nu )$ up to a constant. The constrained mean is ${\tilde{\mu }}_{\nu }=\overline{X}$ for all $\nu$, so its sample-coordinate Jacobian determinant is one, and $j_{\mu \mu }=n/\nu$. The information adjustment givesThis agrees with the exact likelihood based on $Q/\nu \sim {\chi }_{n-1}^2$: estimating the mean has used one degree of freedom. In this setting the Cox-Reid adjusted profile likelihood has the same form. The exact distribution follows by projecting the centered normal vector onto the $(n-1)$-dimensional subspace orthogonal to the constant vector; an orthonormal basis there supplies $n-1$ independent standard normal coordinates whose squares sum to $Q/\nu$.

The example shows both the benefit and the limits. Orthogonality organizes first-order precision and nuisance stability; it does not eliminate all higher-order nuisance effects. Nor does it identify an ancillary statistic or justify dropping the sample-coordinate factor in every modified profile likelihood. In the exponential regression calculation that factor cancels the information adjustment, so an information-only correction would give a different answer.

For a regular two-parameter exponential family, condition on the sample mean of the nuisance canonical statistic. Exponential tilting to its observed value and the saddlepoint density approximation give a nuisance-statistic density proportional to $d_{\lambda \lambda }^{-1/2}$ times its large-deviation exponential. Dividing the joint likelihood by this density gives the profile log-likelihood plus the displayed adjustment, up to a data-only constant. The sign is positive. This construction assumes an interior saddle with positive curvature and a sufficiently regular nonlattice density; conditioning on a lattice statistic requires the corresponding mass approximation.