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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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