Optimal functional expansion for estimation from counting observations

A systematic method to develop approximate nonlinear estimators is presented, in the form of a functional series, for the signal that modulates the rate of a counting process. The estimators are optimal for the given structure and approach the minimum variance (MV) estimator when the approximation order increases. Two kinds of functional series, the iterated integral (II) series and the Fourier-Charlier (FC) series, are used. Product-to-sum formulas for the II and FC functionals are derived. By using the formulas, the MV estimate is projected onto the Hilbert subspaces of the II and the FC series driven by the counting observations with the given index set. The projection results in a Wiener-Hopf type equation for the II kernels and a system of linear algebraic equations for the FC coefficients. The FC series estimator consists of finitely many single Wiener integrals of the counting observations and a nonlinear postprocessor. The nonlinear postprocessor, however, is not memoryless.