Semirecursive nonparametric algorithms for Hammerstein systems with stochastic autocorrelated input

Abstract

A Hammerstein system comprises a nonlinear static subsystem and a linear dynamic subsystem. Herein, semirecursive nonparametric estimators are proposed for the nonlinear static subsystem, and its asymptotic unbiasedness and consistency properties are demonstrated. The estimators are competitive in terms of computational cost and data storage capacity. The performance of the proposed algorithms was examined through both Monte Carlo simulation and application to empirical data.

Keywords:

• Semirecursive
• nonlinearity
• asymptotic unbiasedness
• consistency
• two-stage optimisation
• ARX

Data availability statement

The data that support the findings of this study are available from SASHELP library of SAS Institute Inc.. Restrictions apply to the availability of these data, which were used under license for this study. Data are available from the authors with the permission of SAS Institute Inc.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Lemma 7 of Greblicki (2004): Let $|\epsilon |<1$ and ${\alpha }_n\to 0$ as $n\to \mathrm{\infty }$. Then, $\sum _{i=1}^n{\alpha }_i{\epsilon }^{n-i}\to 0$ as $n\to \mathrm{\infty }$.

2 Lemma 8 of Greblicki (2004): Let $|\epsilon |<1$ and $\alpha >0$. Then, (A28) $\sum _{i=1}^n\frac{1}{i^{\alpha }}{\epsilon }^{n-i}\le \frac{c}{n^{\alpha }}$(A28) for some c.

Additional information

Notes on contributors

Ly-Inn Chung

Ly-Inn Chung received her Bachelor degree from National Chung Hsing University (Taipei, Taiwan) in 1983, Master from the Ohio State University (OH, USA) in 1984, and Ph.D. from University of Illinois at Urbana-Champaign (IL, USA) in 1989. All degrees are in Statistics. She worked in the industry as an actuarial consultant in The Wyatt Company (Chicago, Illinois) and a biostatistician in the pharmaceutical company of Abbott Laboratories (Chicago, Illinois) during 1989-1995. Ly-Inn was an Associate Professor in both Yuan-Ze University (Chung-Li, Taiwan) during 1996 to 2003, and National Taipei University (New Taipei City, Taiwan) since 2004. Her current research interests include Nonparametric Statistics, Medical Statistics, Risk Management, Marketing Research, and Machine Learning. She has published papers in Journal of American Statistical Association, Journal of Affective Disorders, Quality of Life Research, Psychiatry Research, Journal of Nervous and Mental Disease, Occupational Therapy International, Journal of Rehabilitation Medicine, etc.

Tsair-Chuan Lin

Tsair-Chuan Lin obtained an M.S. in 1992 and a Ph.D. in 1996, both in statistics, from the Northern Illinois University (DeKalb, Illinois, U.S.A.). He was an Associate Professor in Chang Gung University in 1997 to 2001, first as an Assistant Professor, then as an Associate Professor. Thereafter, he has been with the Department of Statistics in the National Taipei University (New Taipei City, Taiwan) as an Associate Professor in 2001-2017 and a Professor since 2017. His research lies in the general field of time series, specifically cluster analysis and the nonparametric estimation of nonlinear regression models. He has published in the Journal of the Royal Statistical Society, the Journal of Time Series Analysis, Journal of Acoustical Society of America, Signal Processing, IET Signal Processing, IEEE Transactions on Aerospace and Electronic Systems, among others.

Chun-Chao Wang

Chun-Chao Wang received his Ph.D. degree from Vanderbilt University (Tennessee, USA) in 1992. He joined the Department of Statistics, National Taipei University as an Assistant Professor in 1998. He is currently an Associate Professor and specializes in statistical computations especially in Statistical software development, Biostatistics, and Financial engineering.