Robust trimmed regression for heavy-tailed stable data: Competing methods and order statistics

Document Type : Original Article

Authors

Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Iran

Abstract

Robust regression methods including, least trimmed squares, are among the most important methodologies for computing exact coefficient estimators when data is polluted with outliers. There is interest in generalizing least trimmed squares for regression models with heavy-tailed stable errors. This manuscript, compares estimating coefficients methods with the robust least trimmed squares method in stable errors case. Therefore, we propose stable least trimmed squares and nonlinear stable least trimmed squares methods for linear/nonlinear regression models with stable errors, respectively. The joint distribution of ordered errors is used with the finite variance property of ordered stable errors, whose indexes are defined by cut-off points (Subsection 3.1). We make many comparisons using simulated and real datasets.

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References

[1] R. J. Adler, R. E. Feldman, and M. S. Taqqu, eds., A practical guide to heavy tails, Birkh¨auser Boston, Inc., Boston, MA, 1998. Statistical techniques and applications, Papers from the workshop held in Santa Barbara, CA, December 1995.
[2] M. B. S. Albasatneh and A. Mohammadpour, Least trimmed squares for regression models with stable errors, Fluctuation and Noise Letters, 22 (2023), p. 2350049.
[3] R. Blattberg and T. Sargent, Regression with non-Gaussian stable disturbances: Some sampling results, Econometrica, 39 (1971), pp. 501–510.
[4] M. J. Box, D. Davies, and W. H. Swann, Non-linear Optimization Techniques, Mathematical and Statistical Techniques for Industry, Monograph no. 5, Oliver and Boyd, 1969. For Imperial Chemical Industries.
[5] D. Chwirut, Ultrasonic reference block study, 1979. National Institute of Standards and Technology.
[6] P. L. Davies and U. Gather, Breakdown and groups, Ann. Stat., 33 (2005), pp. 977–1035.
[7] A. C. Davison and D. V. Hinkley, Bootstrap methods and their application, vol. 1 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 1997.
[8] D. Donoho and P. J. Huber, The notion of breakdown point, in A Festschrift for Erich L. Lehmann, Wadsworth Statist./Probab. Ser., Wadsworth, Belmont, CA, 1983, pp. 157–184.
[9] S. Dorniani, A. Mohammadpour, and N. Nematollahi, Ranked set sampling method for estimation of linear regression model with stable errors, in The Eighth World Congress on Probability and Statistics, Istanbul, Turkey, 2012.
[10] H. El Barmi and P. I. Nelson, Inference from stable distributions, in Selected Proceedings of the Symposium on Estimating Functions (Athens, GA, 1996), vol. 32 of IMS Lecture Notes-Monograph Series, Inst. Math. Statist., Hayward, CA, 1997, pp. 439–456.
[11] E. F. Fama, Mandelbrot and the stable paretian hypothesis, The Journal of Business, 36 (1963), pp. 420–429.
[12] A. G. Glen, Maximum likelihood estimation using probability density functions of order statistics, Computers & Industrial Engineering, 58 (2010), pp. 658–662.
[13] F. R. Hampel, A general qualitative definition of robustness, Ann. Math. Stat., 42 (1971), pp. 1887–1896.
[14] P. J. Huber, Robust statistics, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1981.
[15] R. Koenker, Quantile regression, vol. 38 of Econometric Society Monographs, Cambridge University Press, Cambridge, 2005.
[16] J. H. McCulloch, Linear regression with stable disturbances, in A Practical Guide to Heavy Tails: Statistical Techniques and Applications, R. J. Adler, R. Feldman, and M. Taqqu, eds., Boston: Birkh¨auser, 1998, pp. 359– 376.
[17] L. Mili, V. Phaniraj, and P. J. Rousseeuw, Least median of squares estimation in power systems, IEEE Transactions on Power Systems, 6 (1991), pp. 511–523.
[18] M. Mohammadi and A. Mohammadpour, Estimating the parameters of an α-stable distribution using the existence of moments of order statistics, Stat. Probab. Lett., 90 (2014), pp. 78–84.
[19] J. P. Nolan, Maximum likelihood estimation and diagnostics for stable distributions, in L´evy processes, Birkh¨auser Boston, Boston, MA, 2001, pp. 379–400.
[20] , Univariate stable distributions: models for heavy tailed data, Springer Series in Operations Research and Financial Engineering, Springer, Cham, 2020.
[21] J. P. Nolan and D. Ojeda-Revah, Linear and nonlinear regression with stable errors, J. Econometrics, 172 (2013), pp. 186–194.
[22] R. A. Noughabi and A. Mohammadpour, Regression with stable errors based on order statistics, Fluctuation and Noise Letters, 21 (2022), p. 2250014.
[23] H. Riazoshams, H. Midi, and G. Ghilagaber, Robust nonlinear regression: with applications using R, John Wiley & Sons, Inc., Hoboken, NJ, 2019.
[24] P. J. Rousseeuw, Least median of squares regression, J. Amer. Statist. Assoc., 79 (1984), pp. 871–880.
[25] , Multivariate estimation with high breakdown point, Mathematical statistics and applications, (1985), pp. 283–297.
[26] , Introduction to positive-breakdown methods, in Robust inference, vol. 15 of Handbook of Statist., NorthHolland, Amsterdam, 1997, pp. 101–121.
[27] P. J. Rousseeuw and K. V. Driessen, Computing LTS regression for large data sets, Data Min. Knowl. Discov., 12 (2006), pp. 29–45.
[28] P. J. Rousseeuw and A. M. Leroy, Robust regression and outlier detection, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Inc., New York, 1987.
[29] A. R. Soltani and A. Shirvani, Truncated stable random variables: characterization and simulation, Comput. Statist., 25 (2010), pp. 155–161.
[30] W. Stahel and S. Weisberg, eds., Directions in robust statistics and diagnostics. Part II. Proceedings of the IMA 1989 summer program ’Robustness, diagnostics, computing and graphics in statistics’, Minneapolis, MN (USA), vol. 34 of IMA Vol. Math. Appl., Springer New York, NY, 1991.
[31] A. J. Stromberg and D. Ruppert, Breakdown in nonlinear regression, J. Amer. Statist. Assoc., 87 (1992), pp. 991–997.
 [32] M. Teimouri and S. Nadarajah, On simulating truncated stable random variables, Comput. Statist., 28 (2013), pp. 2367–2377.
[33] P. Cˇ´ıˇzek, Least trimmed squares in nonlinear regression under dependence, J. Statist. Plann. Inference, 136 (2006), pp. 3967–3988.
AUT Journal of Mathematics and Computing
Volume 6, Issue 3
July 2025
Pages 241-255

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