On Optimal Uniform Deconvolution

This paper concerns the nonstandard problem of uniform deconvolution for nonperiodic functions over the real line. New algorithms are developed for this nonstandard statistical problem and integrated mean squared error bounds are established. We show that the upper bound of the integrated mean squared error for our new procedure is the same as for the standard case; hence these new estimators attain the lower bound minimax, and hence optimal, rate of convergence. Our method has potential applications to such problems as the deblurring of optical images which have been subjected to uniform motion over a finite interval of time. We also treat the case when the support of the uniform is not given and must be estimated. The numerical properties of our algorithms are demonstrated and shown to be well behaved. AMS Subject Classification: Primary 62G20. Secondary 65R32. Key-words: Deconvolution; characteristic functions; Fourier inversion; rate optimality; uniform distribution.