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Real Time Analysis Based on Reproducing Kernel Henderson Filters

Recently, reproducing kernel Hilbert spaces have been introduced to provide a common approach for studying several nonparametric estimators used for smoothing functional time series data (Dagum and Bianconcini, 2006 and 2008). The reproducing kernel representation is based on the derivation of the density function (i.e. a second order kernel) embedded on the linear filter. This is the starting point for deriving higher order kernels, which are obtained from the product of the density and its orthonormal polynomials. This paper focuses on the Henderson filter, for which two density functions and corresponding hierarchies have been derived. The properties of the Henderson reproducing kernels are analyzed when the filters are adapted at the end of the sample period. The optimality criterion satisfied as well as the influence of the kernel order and bandwidth parameter are studied.