Mathematics – Statistics Theory
Scientific paper
2010-09-21
Mathematics
Statistics Theory
Scientific paper
This paper considers convolution equations that arise from problems such as deconvolution and non-parametric regression with errors in variables. The equations are examined in spaces of generalized functions to account for possible singularities; identification is proved for a wide class of problems. Conditions for well-posedness in the topology of generalized functions are derived for the deconvolution problem and some regressions; an example shows that even in this weak topology well-posedness may not hold. Well-posedness is crucial for consistency of non-parametric deconvolution and important in cases when a non-parametric model is mis-specified as parametric. Stochastic properties and convergence for generalized random processes are derived for solutions of convolution equations. This paper focuses on independent data. Deconvolution estimation for a generalized function on a bounded support in a fixed design case is examined; the rate for a shrinkage deconvolution estimator useful for a sparse support is derived. A consistent non-parametric estimator in an errors in variables regression model with the regression function in L1 is constructed.
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