A lower bound on the error in nonparametric regression type problems. (English) Zbl 0651.62028

Yatracos, Yannis G.

Ann. Stat. 16, No. 3, 1180-1187 (1988).

Let \((X_ 1,Y_ 1),...,(X_ n,Y_ n)\) be a sample, denote the conditional density of \(Y_ i| X_ i=x_ i\) as \(f(y| x_ i,\theta (x_ i))\) and \(\theta\) an element of a metric space (\(\Theta\),d). A lower bound is provided for the d-error in estimating \(\theta\). The order of the bound depends on the local behavior of the Kullback information of the conditional density. As an application, we consider the case where \(\Theta\) is the space of q-smooth functions on \([0,1]^ d\) metrized with the \(L_ r\) distance, \(1\leq r<\infty\).

MSC:

62G05	Nonparametric estimation
62C20	Minimax procedures in statistical decision theory

Keywords:

nonparametric regression; lower bound on minimax risk; lower bound of loss in probability; optimal rates of convergence; conditional density; local behavior of the Kullback information