 
Summary: Wavelet Decomposition Approaches
to Statistical Inverse Problems
BY F. ABRAMOVICH
Department of Statistics & Operations Research,
Raymond & Beverly Sackler Faculty of Exact Sciences,
University of Tel Aviv, Ramat Aviv 69978, Israel.
AND B. W. SILVERMAN
Department of Mathematics, University of Bristol, Bristol BS8 1TW, UK.
Summary
A wide variety of scientific settings involve indirect noisy measurements where one
faces a linear inverse problem in the presence of noise. Primary interest is in some
function but data are accessible only about some linear transform corrupted
by noise. The usual linear methods for such inverse problems do not perform satis
factorily when is spatially inhomogeneous. One existing nonlinear alternative
is the waveletvaguelette decomposition method, based on the expansion of the
unknown in wavelet series. In the vaguelettewavelet decomposition method
proposed here, the observed data are expanded directly in wavelet series. The per
formances of various methods are compared through exact risk calculations, in the
context of the estimation of the derivative of a function observed subject to noise.
A result is proved demonstrating that, with a suitable universal threshold some
