Conservative confidence bands in curvilinear regression
Consider the regression model of full rank in which we observe \[ Y_ i=\sum^{k}_{j=1}b_ jf_ j(x_ i)+e_ i,\quad i=1,...,n, \] where \(b_ j\) are unknown, the \(f_ j\) are known, the points \(x_ i\) are known, the \(e_ i\) are i.i.d. \(N(0,\sigma^ 2)\). Let \(I\subset {\mathbb{R}}\) be a closed interval, \(f(x)=(f_ 1(x),...,f_ k(x))'\) defines a function from I to \({\mathbb{R}}^ k\), \^b\(\sim N_ k(b,\sigma^ 2\Sigma)\) and \(s^ 2\) are the least squares estimator of \(b=(b_ 1,...,b_ k)'\) and an unbiased estimator of \(\sigma^ 2\) respectively. The paper gives a method for constructing a simultaneous confidence band for the regression function b'f(x) for \(x\in I\). The author considers Scheffe-type bands, that is, bands of the form \^b'f(x)\(\pm csp(x)\) for \(x\in I\), where \(p(x)=\{f(x)'\Sigma f(x)\}^{1/2}\) and \(c\geq 0\). The method is based on a geometric inequality giving an upper bound for the uniform measure of the set of piecewise differentiable paths with finite length in \(S^{k-1}\), the unit sphere in \({\mathbb{R}}^ k\).
scientific article; zbMATH DE number 3982319