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Adaptive trimmed likelihood estimation in regression

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EN
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EN
In this paper we derive an asymptotic normality result for an adaptive trimmed likelihood estimator of regression starting from initial high breakdownpoint robust regression estimates. The approach leads to quickly and easily computed robust and efficient estimates for regression. A highlight of the method is that it tends automatically in one algorithm to expose the outliers and give least squares estimates with the outliers removed. The idea is to begin with a rapidly computed consistent robust estimator such as the least median of squares (LMS) or least trimmed squares (LTS) or for example the more recent MM estimators of Yohai. Such estimators are now standard in statistics computing packages, for example as in SPLUS or R. In addition to the asymptotics we provide data analyses supporting the new adaptive approach. This approach appears to work well on a number of data sets and is quicker than the related brute force adaptive regression approach described in Clarke (2000). This current approach builds on the work of Bednarski and Clarke (2002) which considered the asymptotics for the location estimator only.
Twórcy
  • Institute of Economic Sciences, Faculty of Law, Administration and Economy Wrocław University, Uniwersytecka 22/26, 50-145 Wroclaw, Poland
  • Mathematics and Statistics, School of Chemical and Mathematical Sciences, Faculty of Minerals and Energy, Murdoch University, Murdoch, WA 6150, Australia
  • Mathematics and Statistics, School of Chemical and Mathematical Sciences, Faculty of Minerals and Energy, Murdoch University, Murdoch, WA 6150, Australia
Bibliografia
  • [1] T. Bednarski and B.R. Clarke, Trimmed likelihood estimation of location and scale of the normal distribution, Austral. J. Statist. 35 (1993), 141-153.
  • [2] T. Bednarski and B.R. Clarke, Asymptotics for an adaptive trimmed likelihood location estimator, Statistics 36 (2002), 1-8.
  • [3] P. Billingsley, Convergence of Probability Measures. New York: John Wiley 1968.
  • [4] R.W. Butler, Nonparametric interval and point prediction using data trimmed by a Grubbs-type outlier rule, Ann. Statist. 10 (1982), 197-204
  • [5] B.R. Clarke, Empirical evidence for adaptive confidence intervals and identification of outliers using methods of trimming, Austral. J. Statist. 36 (1994), 45-58.
  • [6] B.R. Clarke, An adaptive method of estimation and outlier detection in regression applicable for small to moderate sample sizes, Discussiones Mathematicae, Probability and Statistics 20 (2000), 25-50.
  • [7] Y. Dodge and J. Jurečková, Adaptive combination of least squares and least absolute deviation estimators, in: Statistical Data Analysis Based on the L₁-Norm and Related Methods ed. Y. Dodge, North Holland, Elsevier Science Publishers 1987.
  • [8] Y. Dodge and J. Jurečková, Adaptive choice of trimming proportion in trimmed least-squares estimation, Statistics & Probability Letters 33 (1997), 167-176.
  • [9] Y. Dodge and J. Jurečková, Adaptive Regression, New York, Springer-Verlag 2000.
  • [10] D. Gamble, The Analysis of Contaminated Tidal Data, PhD Thesis, Murdoch University, Western Australia 1999.
  • [11] F.R. Hampel, E.M. Ronchetti, P.J. Rousseeuw and W.A. Stahel, Robust Statistics: The Approach Based on Influence Functions, New York, Wiley 1986.
  • [12] P.J. Huber, The behaviour of maximum likelihood estimates under nonstandard conditions, Proc. Fifth Berkeley Symposium on Mathematical Statistics and Probability 1 (1967), 73-101.
  • [13] P.J. Huber, Robust Statistical Procedures, 2nd ed., CBMS-NSF, Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia 1996.
  • [14] L.A. Jaeckel, Some flexible estimates of location, Ann. Math. Statist. 42 (1971), 1540-1552.
  • [15] J. Jurečková, R. Koenker and A.H. Welsh, Adaptive choice of trimming proportions, Ann. Inst. Statist. Math., 46 (1994), 737-755.
  • [16] N.M. Neykov and C.H. Müller, Breakdown point and computation of trimmed likelihood estimators in generalized linear models, Developments in robust statistics (Vorau 2001), Heidelberg, Physica 2003.
  • [17] C.H. Müller and N. Neykov, Breakdown points of trimmed likelihood estimators and related estimators in generalized linear models, J. Statist. Plann. Inference 116 (2004), 503-519.
  • [18] P.J. Rousseeuw, Multivariate estimation with high breakdown point, pp. 283-297 in: Mathematical Statistics and Applications. Vol.B, (1985), eds. Grossman, Pflug, Vincze and Wertz, Dordrecht:Reidel Publishing Co 1983.
  • [19] P.J. Rousseeuw, Least Median of Squares Regression, Journal of the American Statistical Association 79 (1984), 871-880.
  • [20] P.J. Rousseeuw and A.M. Leroy, Robust Regression and Outlier Detection. New York, Wiley 1987.
  • [21] D.L. Vandev and N.M. Neykov, Robust maximum likelihood in the Gaussian case, p. 259-264 in: New Directions in Data Analysis and Robustness, Morgenthaler, S., Ronchetti, E. and Stahel, W.A. (eds.), Birkhäuser Verlag, Basel 1993.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1128
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