Likelihood based inference in non-linear regression models using \(p^*\) and \(R^*\) approach (Q2771548)

A nonlinear regression model \(Y_j=f_j(\bar\vartheta)+\varepsilon_j\), \(j=1,\dots,n\), is considered, where \(Y_j\) are observations, \(f_j\) are known functions, \(\bar\vartheta\in\Theta\subseteq R^m\) is the vector of unknown parameters, and \(\varepsilon_j\) are normally distributed with mean zero and known diagonal variance matrix. Asymptotic approximations for the conditional distribution of the maximum likelihood estimator given and affine ancillary statistics are derived via Barndorff-Nielsen \(p^*\) and \(R^*\) approach. These results are used to adjustment of the signed log-likelihood ratio statistics. This adjustment allows to derive statistics with distributions for which the normal approximation has relative error \(O(n^{-1})\). The results are applied to a model with replicated measurements.