Adjustment of the profile likelihood for a class of normal regression models (Q2742765)

The problem of estimation of parameters of interest in the presence of nuisance parameters is investigated. Let the distribution of the random variable \(Y\) depend on a parameter \(\theta' =(\psi', \lambda')\), where \(\psi\) is the parameter of interest and \(\lambda \in R^{p}\) is the nuisance parameter. Let \(y' = (y_1, \ldots, y_n)\) be a vector of observations on \(Y\) and let \(l(\theta) = l(\psi, \lambda)\) be the full log-likelihood. Let \({\hat\theta}' = ({\hat\psi}', {\hat\lambda}')\) be the maximum likelihood estimate of \(\theta\) and let \({\hat\lambda}_\psi\) be the maximum likelihood estimate of the nuisance parameter \(\lambda\) for a fixed value \(\psi\) of the parameter of interest. The profile log-likelihood \(l_p(\psi)\) is defined by \(l_p(\psi) = l(\psi, {\hat\lambda}_\psi)\) and the profile log-likelihood score function, \(U(\psi)\), by \(U(\psi)=(\partial/\partial \psi)l_p(\psi).\)NEWLINENEWLINENEWLINEHere, the authors consider the class of nonlinear normal regression models which are defined by \(Y \sim N(X(\psi) \lambda, \Sigma(\psi)).\) For this class under some assumptions formulas are presented for \(U(\psi)-m(\psi)\), where \(m(\psi)=E(\partial l_p/\partial \psi)\), and \(l_{ap}(\psi).\) The case \(Y =B\eta +T\tau +\beta NY +Z\), where \(Y\) is the vector of yields, \(\eta\) is the vector of block effects with incidence matrix \(B\), \(\tau\) is the vector of treatment effects with design matrix \(T\), \(N\) is a neighbour matrix, \(\beta\) is the competition coefficient and \(Z\) is a vector of independent \(N(0, \sigma^2)\) variables, is considered in more detail.