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An interest-respecting reparametrization preserves the parameter of interest, or changes it only by a one-to-one function of itself. The nuisance parameter may depend on both old parameters, provided the complete transformation is smoothly invertible. Thus the level sets representing the inferential target are unchanged. Orthogonality can be sought by choosing the new nuisance coordinate along solutions of the orthogonalization equation for two statistical parameters.

For the positive-shape generalized Pareto distribution with shape $\psi >0$ and scale $\sigma >0$, the expected Fisher information matrix for one observation isThe orthogonalization equation for two statistical parameters gives ${\partial }_{\psi }\sigma =-\sigma /(1+\psi )$ at fixed new nuisance coordinate. Thus $\eta =\sigma (1+\psi )$ is a global interest-respecting reparametrization. In coordinates $(\psi ,\eta )$, the information is diagonal with entries $(1+\psi {)}^{-2}$ and $[{\eta }^2(1+2\psi ){]}^{-1}$.

For a regular two-parameter model, write $\psi$ for the parameter of interest and $\lambda$ for the nuisance parameter. If $u_{\psi },u_{\lambda }$ are its score functions, the parameters are orthogonal statistical parameters when the cross entry of the expected Fisher information matrix vanishes:This condition may hold at a particular parameter value or throughout the parameter domain. It concerns the information geometry, not exact independence of finite-sample estimators. Under regular asymptotic efficiency, a diagonal information matrix gives zero first-order asymptotic covariance.

An interest-respecting reparametrization leaves $\psi$ unchanged, or replaces it by a one-to-one function of $\psi$ alone, while changing the nuisance coordinate by a smooth invertible transformation which may depend on both old parameters. Thus it preserves the level sets of the inferential target. Without loss of generality, keep $\psi$ itself and write the old nuisance as $\lambda =g(\psi ,\eta )$, where $\eta$ is the new nuisance coordinate and $g_{\eta }\ne 0$.

The chain rule for the log-likelihood gives new score functionsTheir expected cross product isSetting this to zero, with $I_{\lambda \lambda }>0$, gives the orthogonalization equation for two statistical parametersSolve this ordinary differential equation along curves of constant $\eta$, using $\eta$ to label distinct initial conditions. Smooth local solutions with nonzero transverse derivative produce an invertible interest-respecting reparametrization. This informal construction identifies precisely which derivative is held at fixed new nuisance coordinate.

For the given positive-shape generalized Pareto distribution, its one-observation log-likelihood isDefine $L=\log (1+\psi Y/\sigma )$ and $D=e^{-L}$. The density transformation $y=\sigma (e^L-1)/\psi$, $dy/dL=\sigma e^L/\psi$, givesThus $L$ has an exponential distribution of mean $\psi$. Direct differentiation of the original log-likelihood, holding the other original parameter fixed, givesThe following exponential integrals determine their expected products:For example, $\mathbb{E}{e}^{-kL}=(1+k\psi {)}^{-1}$ follows by integrating the exponential density, and differentiating this integral in $k$ gives $\mathbb{E}(L{e}^{-kL})=\psi /(1+k\psi {)}^2$. These formulas also give $\mathbb{E}{u}_{\psi }=\mathbb{E}{u}_{\sigma }=0$.

For clarity, the nuisance information and cross information can be calculated explicitly. Put $a=1+\psi$. ThenUsing $\mathbb{E}[L-a+aD]=0$, the cross product isThe same moments give $I_{\psi \psi }={\psi }^{-4}\mathbb{E}[L-a+aD{]}^2=2/[(1+\psi )(1+2\psi )]$. Therefore the Fisher information matrix for one observation isIn particular, $I_{\psi \sigma }>0$ for every admissible pair, proving the original shape and scale parameters are not orthogonal. No finite-mean assumption on $Y$ was needed: the transformed logarithm $L$ and the bounded variable $D$ have the required integrable score products even when the Pareto mean is infinite.

The orthogonalization equation for two statistical parameters now becomesIntegration along a constant-$\eta$ curve gives $\sigma (1+\psi )=\eta$. Hence a global interest-respecting reparametrization isIt is smoothly invertible on the entire positive parameter domain, with inverse $\sigma =\eta /(1+\psi )$. The Fisher information matrix transforms as $J^{T}IJ$, whereSubstitution givesThis verifies the orthogonal shape and scale for a generalized Pareto distribution directly, as well as by the differential equation. For an iid sample, every information entry is multiplied by the sample size, so the same parameter orthogonality remains valid.