Heteroscedastic regression models and applications to off-line quality control (Q2771549)

A heteroscedastic regression model is considered where the mean of the response variable is \(\mu(x,\beta)\) and the errors variance is \(\sigma^2(\mu,z,\psi)\), where \(x\) and \(z\) are (maybe overlapping) explanatory variables, and \(\beta\) and \(\psi\) are unknown parameters to be estimated. Note, that \(\sigma\) depends on \(\beta\) only through the mean \(\mu\). A pseudo likelihood estimator is proposed for the estimation of \(\vartheta=(\beta,\psi)\) by \(n\) independent observations, which is a solution of some estimation equation, say \(F(\vartheta)=0\).NEWLINENEWLINENEWLINEA new numerical algorithm is proposed for solving this equation based on the following idea. If \(\vartheta_{(r)}\) is the \(r\)-th approximation for the solution of \(f(\vartheta)=0\), then \(\vartheta_{(r+1)}\) is defined as the solution (in \(\vartheta\)) of \({\mathbf E}(f(\vartheta_{(r)}) |\vartheta)=f(\vartheta_{(r)})\), where \({\mathbf E}(\cdot |\vartheta)\) is the expectation taken under the parameter value \(\vartheta\). Numerical results on comparisons of this algorithm with the Newton-Raphson one are presented. Asymptotic normality of the estimator is demonstrated and asymptotic covariance matrices are evaluated. Applications to off-line quality control problems are considered.