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Adaptively local one-dimensional subproblems with application to a deconvolution problem. (English) Zbl 0785.62038
Let $$X_ 1,\dots,X_ n$$ be i.i.d. random variables with the density function $$f \in {\mathcal F}$$, where $${\mathcal F}$$ is some set of density functions. By a random sample $$X_ 1, \dots,X_ n$$ of the function, $$T\circ f(x)$$ is estimated and the global loss function $L(d,T \circ f)=\left( \int^ b_ a | T \circ f(x)-d(x) |^ pw(x)dx \right)^{1/p}$ is introduced, where $$w(x)$$ is a weight function estimating $$T \circ f(x)$$.
The main result of this paper establishes a class of density functions $${\mathcal F}$$, for which a lower bound holds for the risk function $\min_{\hat T_ n}\sup_{f \in{\mathcal F}}EL^ p(\hat T_ n,T \circ f),$ where $$\hat T_ n$$ is some estimate.
This result is applied to estimation of derivatives of the unknown density function $$(T \circ f=f^{(k)}(x))$$, estimation of convolutions and estimation of regression curves.

##### MSC:
 62G07 Density estimation
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