Second order asymptotics in nonlinear regression
Let \(y_ t=g(x_ t,\beta)+\sigma u_ t\), \(t=1,...,n\), be a nonlinear regression model, where g(\(\cdot,\beta): {\mathcal X}\to R^ 1\) is a known function, \(\beta \in \Xi \subset R^ p\) is an unknown parameter, \(\{u_ t\}_{t=1,2,...}\) are i.i.d. r.v. with \(Eu_ 1=0\), \(Eu^ 2_ 1=1\), \(x_ t\in R^ q\) are nonstochastic known regressors, and \(\sigma\) is an unknown parameter. Under some regularity conditions there exists a positive constant \(c_{{\mathcal K}}>0\) with \[ \sup_{\theta \in {\mathcal K}}P_{\theta}\{\| n^{1/2}({\hat \beta}-\beta)-\sum^{m- 2}_{i=0}n^{-i/2}h_ i\| >c_{{\mathcal K}}n^{-(m-1)/2}(\log n)^{m/2}\}=o(n^{-1/2}), \] where \({\hat \beta}\) is such that \[ Q({\hat \beta})=\min_{\beta \in \Xi^ c}Q(\beta),\quad Q(\beta)=(n^{-1})\sum^{n}_{i=1}(y_ t-g(x_ t,\beta))^ 2, \] \(h_ i(\theta)\) are polynomials of (\(\sigma\) /\(\sqrt{n})g^{i_ 1,...,i_{i+1}}(x_ t,\beta)\) with bounded coefficients for \(\theta\in {\mathcal K}\) and \(n>n_{{\mathcal K}}\) \((f_{\beta}^{i_ 1,...,i_{\ell}}\) denotes the partial derivative \((\partial /\partial \beta_{i_ 1},...,\partial /\partial \beta_{i_{\ell}})f(\beta)\) with \(i_ j\in \{1,...,p\}\), \(j=1,...,l)\). Similar results hold for the least-squares estimator \({\hat \sigma}^ 2\) of a parameter \(\sigma^ 2.\) Second order asymptotically efficient tests for the hypothesis \(H: \gamma\leq \gamma_ 0\) against \(K: \gamma\) \(>\gamma_ 0\) (\(\gamma\) is the first component of \(\beta\) and \(\gamma_ 0\in R^ 1)\) and for the hypothesis \(H: \sigma\) \({}^ 2\leq \sigma^ 2_ 0\) against \(K: \sigma\) \({}^ 2>\sigma^ 2_ 0\), as well as confidence regions for \(\gamma\) and \(\sigma^ 2\) are given.
Second order asymptotically efficient tests
scientific article; zbMATH DE number 4003295
scientific article