Estimation for the nonlinear functional relationship
Let \(\{b_ n\}^{\infty}_{n=1}\) and \(\{a_ n\}^{\infty}_{n=1}\) be sequences of positive real numbers such that \(a_ n\cdot b_ n=n\). Let \[ f(z^ 0_ t,\beta^ 0)=0,\quad t=1,2,...,b_ n \] be a functional relationship, where \(z^ 0_ t\) are unobservable fixed vectors belonging to a parameter space \(\Gamma \subset R^ p\), \(\beta^ 0\in \Omega \subset R^ k\) is a \(1\times k\) vector of unknown parameters, \[ f(z,\beta):\quad \Gamma \times \Omega \to R^ 1. \] The observations are the p-dimensional vectors \[ Z_{nt}=z^ 0_ t+\epsilon_{nt},\quad t=1,...,b_ n, \] where \(\epsilon_{nt}\) are i.i.d. r.v. with mean zero and covariance matrix \(\Sigma_ n=a_ n^{- 1}\Phi\), where \(\Phi >0\) is a fixed matrix. The maximum likelihood estimators \({\hat \beta}\) and \(\hat z_ t\) are the values of \(\beta\) in \(\Omega\) and \(z_ t\) in \(\Gamma\) that minimize \[ \sum^{b_ n}_{t=1}(Z_{nt}-z_ t)\Sigma_ n^{-1}(Z_{nt}-z_ t)' \] subject to \(f(z_ t,\beta)=0\), \(t=1,...,b_ n\). Conditions for consistency and asymptotic normality of \({\hat \beta}\) and \(\hat z_ t\) are investigated. Modifications of the maximum likelihood estimators (bias-adjusted estimators) are also considered.
scientific article; zbMATH DE number 4096602