A note on recursive smoothers for semimartingales

The kernel function method developed by Yamato (1971) to estimate a probability density function essentially is a way of smoothing the empirical distribution function. This paper shows how one can generalize this method to estimate signals for a semimartingale model. A recursive convolution smoothed estimate is used to obtain an absolutely continuous estimate for an absolutely continuous signal of a semimartingale model. It is also shown that the estimator obtained has a smaller asymptotic variance than the one obtained in Thavaneswaran (1988).