Summary
Consider the problem of estimating a regression function by nonparametric regression from a set of data which is contaminated by a long-tailed error distribution. It is well known that nonparametric regression is not robust against outliers in the response variable. Cleveland (1979) proposed robust locally weighted regression, lowess, which has been implemented in S-Plus and is widely used. In lowess, ordinary local regression is used at the first iteration; under heavy contamination, the initial fitting may be too severely distorted so that robust smoothing is not achieved by the iteration. In this paper, we propose a new method for defining robustness weights based on the weighted median distance of the response variables and compare its performance with lowess by a simulation study. It turns out that the proposed method is more appropriate for heavy contamination.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cleveland, W. S. (1979). Robust Locally Weighted Regression and Smoothing Scatterplots. Journal of the American Statistical Association. 74, 829–836
Cleveland, W.S., and Devlin, S.J. (1988). Locally Weighted Regression: An Approach to Regression Analysis by Local Fitting. Journal of the American Statistical Association. 83, 596–610
Fan, J. (1992). Design-adaptive Nonparametric Regression. Journal of the American Statistical Association. 87, 998–1004
Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Chapman & Hall
Härdle, W. (1984). Robust Regression Function Estimation. Journal of Multivariate Analysis. 14, 169–180
Härdle, W. (1990). Applied Nonparametric Regression. Cambridge. Cambridge University Press
Härdle, W., and Gasser, T. (1984). Robust Nonparametric Function Fitting. Journal of the Royal Statistical Society, Ser. B, 46, 42–51
Huber, P.J. (1979). Robust Smoothing, in Robustness in Statistics. eds. R.L. Launer and G.N. Wilkinson. New York. Academic Press
Iman. R.L., and Conover, W.J. (1979). The Use of the Rank Transform in Regression. Technometrics. 21, 499–509
Park, D. (2000). Robust Nonparametic Regression Method using Rank Transformation. The Korean Communications in Statistics. 7, 575–583
Rousseeuw, P.J. and Leroy, A.M. (1987). Robust Regression and Outlier Detection. John Wiley & Sons
Scott, D.W. (1992). Multivariate Density Estimation. New York. John Wiley
Silverman, B.W. (1985). Some Aspects of the Spline Smoothing Approach to Non-Parametric Regression Curve Fitting (with discussion). Journal of the Royal Statistical Society. Ser. B. 47, 1–52
Wagner, H.M. (1959), Linear Programming Techniques for Regression Analysis. Journal of the American Statistical Association. 54, 206–212
Wang, F.T., and Scott, D.W. (1994). The L1 Method for Robust Nonparametric Regression. Journal of the American Statistical Assocation. 89, 65–76
Author information
Authors and Affiliations
Additional information
This research was supported by the Hanshin University Special Research Grants in 2003.
Rights and permissions
About this article
Cite this article
Park, D. Robustness Weight by Weighted Median Distance. CompStat 19, 367–383 (2004). https://doi.org/10.1007/BF03372102
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03372102