Variable bandwidth kernel estimators of regression curves (Q1096989)

Observations \(Y_ 1,...,Y_ n\) are generated according to \[ (1)\;Y_ i=g(t_ i)+ \epsilon_ i,\;i=1,...,n, \] where \(g\in Lip([0,1])\) is the curve to be estimated and \(\epsilon_ 1,...,\epsilon_ n\) are i.i.d. noise variables satisfying \(E(\epsilon_ 1)=0\) and \(E(\epsilon^ 2_ 1)=\sigma^ 2<\infty\). The design \(t_ 1,...,t_ n\) is fixed in advance as \(t_ i=i/n\). Two kernel estimators of g(t) are considered. The first estimator is \[ (2)\;\hat G(t,b)=(1/nb)\sum^{n}_{i=1}K((t- t_ i)/b)Y_ i \] where K is a kernel function and \(b=b(n)\) is a sequence of bandwidths such that \(b\to 0\) and \(nb^ 2\to \infty\) as \(n\to \infty\). The second estimator is \[ (3)\;g(t,b_ t)=(1/nb_ t)\sum^{n}_{i=1}K((t-t_ i)/b_ t)Y_ i \] where \(b_ t\) is a function of t, \(0\leq t\leq 1\). Both b and \(b_ t\) may involve datadriven selections. In this paper, optimality refers to minimization of the leading terms of asymptotic expressions of integrated mean square error (IMSE). Let \(b^*\) and \(b^*_ t\) be the optimal bandwidths for (2) and (3), and let \(\tilde b_ t^*\) be a truncated version of \(b_ t^*\) (there is a cut-off if \(b^*_ t\) is too large). Then \[ \lim_{n\to \infty}n^{2k/(2k+1)}IMSE(\hat g(\cdot,\tilde b_ t^*))\leq \lim_{n\to \infty}n^{2k/(2k+1)}IMSE(\hat G(\cdot,b^*)) \] if \(g\in C^ k([0,1])\) for some \(k\geq 2\) and the kernel function K satisfies \[ (i)\;K\in Lip([-1,1]),\;(ii)\;\int^{1}_{-1}K(x)dx=1, \] and \[ (iii)\;\int^{1}_{-1}x^ jK(x)dx \begin{cases} = 0 &\text{ if \(0<j<k,\)} \\ \neq 0 &\text{ if \(j=k.\)}\end{cases} \] In addition, it is shown that \(\hat g(\cdot,\hat b_ t)\) behaves asymptotically as well as \(\hat g(\cdot,b^*_ t)\) when \(\hat b_ t\) is a consistent estimator of \(b^*_ t\). Finally, for fixed \(t_ 0\in [0,1]\) it is shown that \[ n^{k/(2k+1)}(\hat g(t,\tau n^{-1/(2k+1)})-g(t)) \] converges in distribution to a Gaussian limit process on C([r,s]) for some \(\tau\)- interval [r,s], and hence any consistent estimator for \(b^*_ t\) will be asymptotically efficient.