Abstract
The purpose of the paper is to show that the approach developed in the preceding part can also be used to construct approximate estimators for the signal that modulates the rate of a counting process. The estimators are optimal for the given structure and approach the best estimate, when the approximation order increases.
A product-to-sum formula for the multiple Wiener integrals, with respect to the counting process martingale, is derived. By using the formula, the minimal variance estimate is projected onto the Hilbert subspace of all Fourier-Charlier (FC) series, driven by the counting observations, with the same given index set. The projection results in a system of linear algebraic equations for the FC coefficients, the parameters of the desired approximate estimator.
The estimator consists of finitely many Wiener integrals of the counting observations and nonlinear postprocessor. The non-linear postprocessor, however, is not memoryless. More work is required here.
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© 1983 D. Reidel Publishing Company
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Ting-Ho Lo, J., Sze-Kui, N. (1983). Optimal Orthogonal Expansion for Estimation II: Signal in Counting Observations. In: Bucy, R.S., Moura, J.M.F. (eds) Nonlinear Stochastic Problems. NATO ASI Series, vol 104. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-7142-4_24
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DOI: https://doi.org/10.1007/978-94-009-7142-4_24
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