CROSS-VALIDATION FOR PARAMETER SELECTION IN INVERSE ESTIMATION PROBLEMS

Inverse estimation is concerned with estimation of a signal, where the signal is indirectly observed in the presence of random noise. By ind irect we mean, more precisely, that we observe a transform of the sign al, The transforms that we consider here are quite general and in part icular not restricted to compact operators. This general abstract setu p covers many examples of practical interest, One possible approach to signal recovery is to apply a regularized inverse of the transform to the observed image. Although optimal rates for the smoothing paramete r when the number of data points increases indefinitely can be establi shed, this does not give any information about a suitable choice of th e parameter for a fixed given data set. In order to solve that problem we propose a cross-validation method that can even be formulated in t he general abstract case, where the expression is given in the ''spect ral domain''. In the special instances of indirect density and regress ion estimation, moreover, an equivalent expression in the ''time domai n'' can be given. A simulation study is devoted to a deconvolution pro blem with quite satisfactory results.