Random approximations to some measures of accuracy in nonparametric curve estimation (Q1085912)

Given a random sample \(X_ 1,..,X_ n\) of d-dimensional random vectors having distribution function F(x), a function g(x) (e.g. probability density function, regression function, hazard function) is to be estimated. The class of ''fractional delta sequence estimators'' \[ \hat g(x)=\sum^{n}_{i=1}\delta_{\lambda}(x,X_ i)/\sum^{n}_{i=1}\delta '_{\lambda}(x,X_ i) \] is considered, where \(\delta_{\lambda}\) and \(\delta '_{\lambda}\) are measurable functions on \({\mathbb{R}}^ d\times {\mathbb{R}}^ d\), which are indexed by a ''smoothing parameter'' \(\lambda =\lambda (n)\in {\mathbb{R}}^+\). As a meaasure of accuracy of estimation, \[ MISE=E\int [\hat g(x)-g(x)]^ 2w(x)dF(x), \] where w(x) is a nonnegative weight function, is adopted, and its approximation by \[ ISE=\int [\hat g(x)-g(x)]^ 2w(x)dF(x) \] or by \[ ASE=n^{-1}\sum^{n}_{i=1}[\hat g(X_ i)-g(X_ i)]^ 2w(X_ i) \] is considered. The main results state that \[ \lim_{n\to \infty}\sup_{\lambda}| \frac{ISE-MISE}{MISE}| =0\quad a.s. \] and \[ \lim_{n\to \infty}\sup_{\lambda}| \frac{ASE- MISE}{MISE}| =0\quad a.s. \]