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On the diversity of estimates. (English) Zbl 1052.62509
Summary: The presented examples of regression analysis of some real data sets below confirm that it is possible that various estimators may produce considerably different estimates. Such situation will be called diversity of estimates. Employing high breakdown point estimators, namely the least median of squares (LMS) and the least trimmed squares (LTS), the diversity of estimates is demonstrated. Two examples with the artificial data and a formal reflection of the diversity of estimates hint why the diversity of estimates takes place. The theoretical reflection also shows that the diversity of estimates may appear for any sample size. A proposal an how to select from the diverse estimates of model is given and illustrated in examples of real data sets.

MSC:
62F10 Point estimation
62F99 Parametric inference
62F35 Robustness and adaptive procedures (parametric inference)
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[1] Benáček, V., Jarolı\acutem, M., Vı\acutešek, J.Á., 1998. Supply-side characteristics and the industrial structure of czech foreign trade (together with Prof. Vladimı\acuter J. Benáček a Martin Jarolı\acutem). Proc. Conf. Business and Economic Development in Central and Eastern Europe: Implications for Economic Integration into Wider Europe, ISBN 80-214-1202-X, published by Technical university in Brno together with University of Wisconsin, Whitewaters, and the Notingham Trent university, pp. 51–68.
[2] Benáček, V., Vı\acutešek, J.Á., 1998. Determining factors and effects of foreigner direct investment in an economy transition: evidence from the Czech manufacturing industries in 1991–1997. Preprint.
[3] Boček, P., Lachout, P., 1993. Linear programming approach to LMS-estimation. Memorial volume Comput. Statist. Data Anal. devoted to T. Havránek, to appear.
[4] Carroll, R. J.; Ruppert, D.: Transformations in regression: a robust analysis. Technometrics 27, 1-12 (1985) · Zbl 0558.62032
[5] Chatterjee, S., Hadi, A.S., 1988. Sensitivity Analysis in Linear Regression. Wiley, New York. · Zbl 0648.62066
[6] Chatterjee, S., Price, B., 1977. Regression Analysis by Example. Wiley, New York. · Zbl 0900.62356
[7] Daniel, C., Wood, F.S., 1980. Fitting Equations to Data. Wiley, New York. · Zbl 0418.65006
[8] Davis, P. L.: Aspects of robust linear regression.. Ann. statist. 21, 1843-1899 (1993) · Zbl 0797.62026
[9] Draper, N.R., Smith, H., 1981. Applied Regression Analysis. Wiley, New York. · Zbl 0548.62046
[10] Fisher, R. A.: On the mathematical foundations of theoretical statistics. Philos. trans. Roy. soc. London ser. A 222, 309-368 (1922) · JFM 48.1280.02
[11] Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., Stahel, W.A., 1986. Robust Statistics – The Approach Based on Influence Functions. Wiley, New York. · Zbl 0593.62027
[12] Hettmansperger, T. P.; Sheather, S. J.: A cautionary note on the method of least median squares. The amer. Statistician 46, 79-83 (1992)
[13] Joss, J.; Marazzi, A.: Probabilistic algorithms for LMS regression. Comput. statist. Data anal. 9, 123-134 (1990)
[14] Jurečková, J.; Portnoy, S.: Asymptotics for one-step M-estimators in regression with application to combining efficiency and high breakdown point. Commun. statist. A 16, 2187-2199 (1988) · Zbl 0656.62041
[15] Kovanic, P.: A new theoretical and algorithmic basis for estimation, identification and control. Automatica 22, 657-674 (1986) · Zbl 0608.93002
[16] Markatou, M., Stahel, W.A., Ronchetti, E.M., 1991. Robust M-type testing procedures for linear models. In: Stahel W., Weisberg S. (Eds.), Directions in Robust Statistics and Diagnostics. Springer, New York, pp. 201–220. · Zbl 0735.62035
[17] Maronna, R. A.; Yohai, V. J.: Asymptotic behaviour of general M-estimates for regression and scale with random carriers. Z. wahrscheinlichkeitstheorie verw. Gebiete 58, 7-20 (1981) · Zbl 0451.62031
[18] Martin, R. D.; Yohai, V. J.; Zamar, R. H.: MIN-MAX bias robust regression. Ann. statist. 17, 1608-1630 (1989) · Zbl 0713.62068
[19] Mason, R.L., Gunst, R.F., Hess, J.L., 1989. Statistical Design and Analysis of Experiments. Wiley, New York. · Zbl 1029.62068
[20] Prigogine, I., Stengers, I., 1977. La Nouvelle Alliance. SCIENTIA, 1977, issues 5–12.
[21] Prigogine, I., Stengers, I., 1984. Out of Chaos. William Heinemann Ltd, London, 1984.
[22] Rousseeuw, P. J.: Least median of square regression. J. amer. Statist. assoc. 79, 871-880 (1984) · Zbl 0547.62046
[23] Rousseeuw, P.J., 1993. Unconventional features of positive-breakdown estimators. Report no. 93-26 of University of Antwerp, UIA, Belgium. · Zbl 0791.62036
[24] Rousseeuw, P.J., Leroy, A.M., 1987. Robust Regression and Outlier Detection. Wiley, New York. · Zbl 0711.62030
[25] Rubio, A.M., Quintana, F., Vı\acutešek, J.Á., 1994. Diagnostics of non-linear regression. Trans. 12th Prague Conf. on Information Theory, Statistical Decision Functions and Random Processes, Prague, 1994, pp. 203–206.
[26] Rubio, A.M., Vı\acutešek, J.Á., 1994. Diagnostics of regression model: test of goodness of fit. In: Hušková, M., Mandl, P. (Eds.), Trans. 5th Prague Symp. on Asymptotic Statistics. Springer, Berlin, pp. 425–434.
[27] Ruppert, D.; Carroll, R. J.: Trimmed least squares estimation in linear model. J. amer. Statist. assoc. 75, No. 372, 828-838 (1980) · Zbl 0459.62055
[28] Simpson, D. G.; Ruppert, D.; Carroll, R. J.: On one-step GM estimates and stability of inferences in linear regression. J. amer. Statist. assoc. 87, 439-450 (1992) · Zbl 0781.62104
[29] Vı\acutešek, J.Á., 1994. A cautionary note on the method of least median of squares reconsidered, Trans. 12th Prague Conf. on Information Theory, Statistical Decision Functions and Random Processes, Prague, pp. 254–259.
[30] Vı\acutešek, J.Á., 1996a. On the heuristics of statistical results. Proceedings of ’PROBASTAT’94’, Vol. 7, Tatra Mountains Publ. 1996, pp. 349–357, Bratislava. · Zbl 0925.62020
[31] Vı\acutešek, J. Á: Sensitivity analysis of M-estimates. Ann. ins. Statist. math. 48, 469-495 (1996) · Zbl 0925.62131
[32] Vı\acutešek, J. Á: Diagnostics of regression subsample stability. Probab. math. Statist. 17, No. 2, 231-257 (1997) · Zbl 0924.62072
[33] Vı\acutešek, J.Á., 1999. The least trimmed squares estimator asymptotic representation and sensitivity studies, (preprint).
[34] Yohai, V. J.; Zamar, R. H.: A minimax-bias property of the least \({\alpha}\)-quantile estimates. Ann. statist. 21, 1824-1842 (1993) · Zbl 0797.62027
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