A method for estimating intensity and impulse response functions of filtered Poisson processes

This paper presents a method for estimating the intensity and impulse response (IR) function of filtered Poisson processes. The case is considered where the filtered Poisson process is modeled as an output of the linear constant-coefficient ordinary differential equations having poles and zeros driven by Poisson impulse processes. It is shown that an explicit formula for estimating the intensity is derived by combining second-and third-order cumulants of the residual time series generated from the discretized filtered Poisson process. It is also shown that the IR function can be estimated from the parameters of the discretized filtered Poisson process. Then, Monte Carlo simulations demonstrate the validity of the proposed method in some specific examples. It is concluded that the proposed method can be extensively applied to actual phenomena appearing in engineering and science.