Multiparameter bandwidth processes and adaptive surface smoothing
For the problem \(y_{i,n}= g(x_{i,n})+ \varepsilon_{i,n}\), \(i=1,2,\dots,n\) and \(x_{i,n}\in \mathbb{R}^ m\), and for the nonparametric estimation of the function \(g\) using the kernel method, the authors provide a sequence of window processes that converge towards a Gaussian process, having mean and covariance known. The limit process gives a criterion to choose the optimal window. Their procedure is as follows: a classical analysis of bias and variance shows that the general optimal window is \(b_ i= t_ i n^{- (1/2k+m)}\), \(i=1,\dots,m\), where \(k\) is the smoothness degree of \(g\). Hence it is only necessary to determine the \(t_ i\). Then, they prove: \(\gamma_ n (\widehat{g}^{(\nu)} (x,t)- D^{(\nu)} g(x))\) converges towards \(\eta ({\mathbf t})\), \({\mathbf t}= (t_ 1,\dots,t_ m)\) and \(\eta({\mathbf t})\) a Gaussian process, where \(\widehat{g}^{(\nu)}\) is the estimator of \(D^{(\nu)}g\), \(0<a_ 1\leq t_ i\leq a_ 2<\infty\), and taking as window the \(b_ i\). In particular, when \({\mathbf t}\) is fixed this theorem gives the asymptotic mean quadratic error (AMSE). Theoretical optimal estimators can be obtained minimizing the AMSE and introducing then this value in the estimators.
scientific article; zbMATH DE number 445135