Analysis of the accuracy of the linear regression model for a change in the number of parameters (Q918608)

The author considers the regression model \(Y_ i=f(x_ i)+e_ i\), \(i=1,...,n\), where \(x_ 1,...,x_ n\) are fixed design points, \(e_ 1,...,e_ n\) are independent, identically distributed random errors with zero mean and variance \(\sigma^ 2>0\). The function f is assumed to belong either to \[ {\mathcal F}_ k=\{\sum^{k}_{i=1}c_ i\phi_ i,\quad c_ i\in R_ 1,\quad i=1,...,k\}\quad or\quad to\quad F_{\infty}=\{\sum^{\infty}_{i=1}c_ i\phi_ i,\quad c_ i\in R_ 1,\quad i=1,...,\}, \] where \(\{\phi_ i\}_{i\geq 1}\) is a sequence of given functions (usually linearly independent). The mean square error \[ E\sum^{k}_{i=1}(f(x_ i)-\hat f(x_ i))^ 2,\text{ where } \hat f(x)=\sum^{k}_{i=1}\hat c_ i\phi_ i \] with \((\hat c_ 1,...,\hat c_ k)\) being the least squares estimators of \((c_ 1,..,x_ k)\), k fixed, is investigated.