ISBIS4 Abstract

Contact Author's Name: MD Jankowitz
Title of Abstract: Non-Linear (LULU) smoothers for Time Series Data
Author(s): MD Jankowitz, WJ Conradie and T de Wet
Affiliation: University of South Africa(UNISA), University of Stellenbosch

LULU smoothers is a class of non-linear smoothers that was introduced by Rohwer (1989) and has since been studied extensively by him, from a mathematical point of view, culminating in the publishing of Rohwer (2004). These smoothers are compositions of minima and maxima and have very attractive mathematical properties compared to other non-linear smoothers. They have also been successfully applied in image processing, engineering and the earth sciences.

In this talk LULU smoothers will firstly be introduced and a review given of their attractive mathematical properties. Very little attention has been given to their distributional and statistical properties in the literature. We have recently made some progress with this and will secondly give some new results on the distributions of these smoothers, for the simplest ones as well as for more complex ones in the class. Thirdly the performance of LULU smoothers will be evaluated by comparing them to the median smoother and some of its modifications and extensions.

Fourthly, we will report on some asymptotic results of LULU smoothers when the window size tends to infinity. These limiting distributions are obtained in terms of the class of extreme value distributions. Some numerical and simulation results will also be given. Fifthly the attractive way LULU smoothers deal with impulsive noise in the form of blockpulses and of decomposing the variation in a series will be highlighted and illustrated by applying it to financial time series. Finally some provisional results regarding the application of LULU smoothers in the estimation of volatility will be discussed and compared with existing models, for example ARCH-, GARCH- and Realised Volatility models.

Rohwer, C.H. (1989). Idempotent One-Sided Approximation of Median Smoothers. Journal of Approximation Theory. Vol 58, No 2: 151-163. Rohwer, C.H. (2004). Nonlinear Multiresolution Analysis. Birkhaüser Publishers.