Doubly Smoothed EM Algorithm for Statistical Inverse Problems

Recent results on discretization effects in unfolding intensity function of an indirectly observed nonhomogeneous Poisson process show that a critical feasibility condition for the strong L2 consistency of the histogram sieve maximum likelihood estimates requires that the approximation error of the function of interest with step functions tend to 0 faster than the squared singular values of the discretized folding operator. This condition may not be fulfilled in some standard inverse problems like, for example, the deconvolution problem and the Wicksell problem discretized in a standard way. The condition may be satisfied, however, with suitably modified discrete operator and suitably constructed parametric sets for the discrete problems. Motivated by these results, an additional smoothing step is added to the EMS (expectation-maximization-smoothing) algorithm and an automatic procedure for the choice of a smoothing parameter is proposed. An application to the Wicksell problem is presented. In this context, the theory also suggests a special, nonuniform binning in the data space. A simulation study demonstrates that the new approach considerably reduces the L2 risk when compared to both the standard EMS algorithm and to a spectral procedure specially designed for the Wicksell problem. A priori knowledge of the value of the estimated function at 0 may further reduce the risk very significantly. Theoretical results on the strong L2 consistency and convergence rates are given.