Unfolding intensity function of a Poisson process in models with approximately specified folding operator

Abstract.

Quasi-maximum likelihood histogram sieve estimators of the intensity function of an indirectly observed Poisson process are studied. The setup differs from the standard one in that the exact form of the folding operator may not be known. Instead, approximate knowledge on its discretized version is available. Conditions for strong L ²-consistency are given and admissible discretization rates are studied. In non-folding problems, the number of histogram bins may essentially increase at the usual maximal rate while folding reduces the allowed discretization rates. It is shown that, even in moderately ill-posed problems, the discretization effects may be critical for the strong L ²-convergence and that there is an essential need both for further regularization and for imposing stronger conditions on the estimated function. Not surprisingly, the most restrictive factor is the low approximation power of piecewise constant functions. A regularization method is proposed which suitably modifies the discrete approximation of the folding operator and ensures the strong consistency. Since no penalty term is being introduced, the EM algorithm can be used in its factorized, efficient form. Convergence rates are obtained in terms of the discrete problem.