From exact gradients to exact standard errors: third- and fourth-order sensitivities for the Laplacian population likelihood family

Closed-form FOCE and FOCEI observed information enables exact asymptotic standard errors for nonlinear mixed-effects models, avoiding finite-difference step-size dependence.

• Why it matters: Finite differencing for the Fisher information is less accurate and depends on step size, limiting reliable asymptotic standard errors for FOCE/FOCEI fits.
• What they did: The authors derive FOCE/FOCEI observed information in the same sensitivity-equation framework by splitting the population Hessian into a data term using second-order sensitivities and a log-determinant term requiring third-order sensitivity equations.
• The result: Finite-difference Hessians achieve accuracy no better than the square root of objective-evaluation accuracy, while the analytic form is limited only by sensitivity and differential-equation solution accuracy, implemented in nlmixr2.

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