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