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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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