Nonlinear filters based on taylor series expansions∗

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Abstract
The nonlinear filters based on Taylor series approximation are broadly used for computational simplicity, even though their filtering estimates are clearly biased. In this paper, first, we analyze what is approximated when we apply the expanded nonlinear functions to the standard linear recursive Kalman filter algorithm. Next, since the state variable αt and αt-t are approximated as a conditional normal distribution given information up to time t - 1 (i.e., It-1) in approximation of the Taylor series expansion, it might be appropriate to evaluate each expectation by generating normal random numbers of αt and αt-1 given It-1 and those of the error terms θ and ηt. Thus, we propose the Monte-Carlo simulation filter using normal random draws. Finally we perform two Monte-Carlo experiments, where we obtain the result that the Monte-Carlo simulation filter has a superior performance over the nonlinear filters such as the extended Kalman filter and the second-order nonlinear filter.