1. Founded 1348Charles University http://www.fsv.cuni.cz

2. Institute of Information Theory and Automation Academy of Sciences of the Czech Republ Institute of Information Theory Institute of Economic Studies Faculty of Social Sciences Charles University Prague Institute of Economic Studies Faculty of Social Sciences Charles University Prague http://samba.fsv.cuni.cz/~visek/cox Neuchatel 14. - 18. 7. 2004 LEAST WEIGTED SQUARES FOR DYNAMIC SPECIFICATION and Automation Academy of Sciences of the Czech Republ Jan Ámos Víšek Neuchatel 14. - 18. 7. 2004 LEAST WEIGTED SQUARES FOR DYNAMIC SPECIFICATION http://samba.fsv.cuni.cz/~visek/cox Celebrating Statistics - David Cox

3. Topic of presentation ● Definition of the Least Weighted Squares ● Their properties ● Paradigm of the robust estimation ( which the Least Weighted Squares fulfill) ● Algorithm for their evaluation

4. Consistency Asymptotic normality Reasonably high efficiency Unbiasedness Nearly impossible to fulfill for robust estimators, hence abandoned Nearly “automatically” fulfilled for “classical” estimators, hence frequently unduly ignored in robust regression Bickel, P.J. (1975): One-step Huber estimates in the linear model. Jurečková J., P. K. Sen (1984): On adaptive scale-equivariant M-estimators in linear models. JASA 70, 428-433. Statistics and Decisions, vol. 2 (1984), Suppl. Issue No.1. Requirements on an estimator of regression coefficients naturally inherited from the classical statistics Robust regression E.g. simple M-estimators lack this property, for discussion see Scale- and regression-equivariance

5. Low local shift sensitivity Preferably finite rejection point Quite low gross-error sensitivity Hampel, F. R., E. M. Ronchetti, P. J. Rousseeuw, W. A. Stahel (1986): New York: J.Wiley & Sons. Robust Statistics - The Approach Based on Influence Functions. Requirements on an estimator of regression coefficients naturally stemming from principles of robustness Let’s call these four points Hampel’s paradigm If interested in, ask me for sending by e-mail. Víšek, J. Á. (2003): Development of the Czech export in nineties. In: Consolidation of governing and business in the Czech republic and EU I., 193 - 220, ISBN 80-86732-00-2, MatFyz Press. The applications indicated that “high” should be substituted by “controlable”, see e.g. Robust regression High breakdown point

6. Let us agree, for a while, that the majority of data determines the “true” model. Then a small change even of one observation can cause a large change of estimate. High breakdown point may be sometimes self-destructive Requirements on an estimator of regression coefficients naturally stemming from..... - a comment Robust regression The method too much relies on selected “true” points ! What is the problem ? Hence, it may be preferable to reject observations “smoothly”.

7. Available diagnostics, sensitivity studies and accompanying procedures Existence of an implementation of the algorithm with acceptable complexity and reliability of evaluation If interested in, ask me for sending by e-mail. Víšek, J.Á. (2000): A new paradigm of point estimation. Proc. of Data Analysis 2000/II, Modern Statistical Methods - Modeling, Regression, Classification and Data Mining, ISBN 80-238-6590-0, 195 - 230. Requirements on an estimator of regression coefficients ( nearly) inevitable for successful applications Robust regression Let’s discuss them point by point. An efficient and acceptable heuristics

8. If interested in, ask me for sending by e-mail. Available diagnostics, sensitivity studies and...... Requirements on a robust estimator of regression coefficients ( nearly) inevitable for successful applications Kalina, J. (2003): Autocorrelated disturbances of robust regression. European Young Statistician Meeting 2003 – to appear. Víšek, J.Á. (2003): Durbin-Watson statistic in robust regression. Probability and Mathematical Statistics, vol. 23., Fasc. 2(2003), 435 - 483. Víšek, J.Á. (2002): White test for the least weigthed squares. COMPSTAT 2002, Berlin, Short Communications and Poster (CD), ISBN 3-00-009819-4 (eds. S. Klinke, P. Ahrend, L. Richter). Víšek, J.Á. (2001): Durbin-Watson statistic for the least trimmed squares. Bulletin of the Czech Econometric Society, vol. 8, 14/2001, 1 – 40. Víšek, J.Á. (1998): Robust specification test. Proc. Prague Stochastics'98 (eds. M. Hušková, P. Lachout, Union of Czechoslovak Mathematicians and Physicists), 1998, 581 - 586. Víšek, J.Á. (2003): Estimating contamination level. Proc. Fifth Pannonian Sympos.on Math. Statist., Visegrad, Hungary 1985, 401--414. as or

9. Available diagnostics, sensitivity studies and accompanying procedures Requirements on a robust estimator of regression coefficients ( nearly) inevitable for successful applications Víšek, J.Á. (2002): Sensitivity analysis of M-estimates of nonlinear regression model: Influence of data subsets. Ann. Inst.Statist. Math., 54, 2, 261 - 290. Víšek, J.Á. (1997): Contamination level and sensitivity of robust tests. Handbook of Statist. 15, 633 – 642 (eds. G. S. Maddala & C. R.. Rao) Amsterdam: Elsevier Science B. V. Víšek, J.Á. (1996): Sensitivity analysis of M-estimates. Ann. Inst.Statist. Math., 48(1996), 469-495. If interested in, ask me for reprints. Víšek, J.Á. (1986): Sensitivity of the test error probabilities with respect to the level of contamination in general model of contaminacy. J. Statist.Planning and Inference 14,(1986), 281--299. Jurečková J., J. Á. Víšek (1984): Sensitivity of Chow--Robbins procedure to the contamination. Commun. Statist. -- Sequential Analys. 1984 3 (2), 175--190. or as

10. Available diagnostics, sensitivity studies and accompanying procedures Requirements on a robust estimator of regression coefficients ( nearly) inevitable for successful applications If interested in, ask me for sending by e-mail. Víšek, J.Á. (1997): Robustifying instrumental variables. Submitted to COMPSTAT 2004. Víšek, J.Á. (1996): Selecting regression model. Probability and Mathematical Statistics 21,. 2 (2001), 467 – 492. Víšek, J.Á. (2000): Robust instrumental variables and specification test. Proc. PRASTAN 2000, ISBN 80-227-1486-0, 133 - 164.. Víšek, J.Á. (1998): Robust instruments. Proc. Robust'98 (ed. J. Antoch & G. Dohnal) Union of Czechoslovak Mathematicians and Physicists, 195 - 224. as or

11. Existence of an implementation of the algorithm with acceptable complexity and reliability of evaluation Requirements on a robust estimator of regression coefficients ( nearly) inevitable for successful applications Hettmansperger, T.P., S. J. Sheather (1992): A Cautionary Note on the Method of Least Median Squares. The American Statistician 46, 79-83. - the timing of sparks - air / fuel ratio - intake temperature - exhaust temperature Explanatory variables: Response variable: Number of knocks of an engine Number of observations: 16 Engine knock data - treated by the Least Median of Squares The results were due to bad algorithm, they used. They are on the next page. A small change (7.2%) of one value in data caused a large change of the estimates.

12. Existence of an implementation of the algorithm with.... Requirements on a robust estimator of regression coefficients ( nearly) inevitable for successful applications DataIntrc.sparkairintakeexhaust11 th res. Correct30.080.212.900.560.930.570 Wrong-86.54.591.211.47.0690.328 Engine knock data - results by Hettmansperger and Sheather DataIntrc.sparkairintakeexhaust11 th res. Correct30.040.143.080.46-.0070.450 Wrong48.38-.733.390.19-.0110.203 Boček, P., P. Lachout (1995): Linear programming approach to LMS-estimation. Mem. vol. Comput. Statist. & Data Analysis 19 (1995), 129 - 134.. A new algorithm, based on simplex method, was nearly immediately available, although published a bit later. It indicates that the reliability of algorithm and its implementation is crucial. Minimized squared residual

13. An efficient and acceptable heuristics (?) Requirements on a robust estimator of regression coefficients ( nearly) inevitable for successful applications hints that, in the case of sufficient “demand for data-processing”, we may “cope” without any heuristics. - it seems quit acceptable heuristics, unfortunately it does not work, - for the example of data for which the min-max-estimator failed see - maximum was taken over some set of underlying d.f.’s and minimum over possible estimators, Víšek, J.Á. (2000): On the diversity of estimates. CSDA 34, (2000) 67 - 89. But papers like -the problem is that the method implicitly takes maximum over “unexpected” set of d.f.’s. Hansen, L. P. (1982): Large sample properties of generalized method of moments estimators. Econometrica, 50, no 4, 1029 - 1054. In 1989 Martin et al. studied estimators minimizing maximal bias of them Martin, R.. D., V. J. Yohai, R. H. Zamar (1989): Min-max bias robust regression. Ann Statist. 17, 1608 - 1630.

14. non-increasing, absolutely continuous If interested in, ask me for sending by e-mail. Víšek, J.Á. (2000): Regression with high breakdown point. ROBUST 2000, 324 – 356, ISBN 80-7015-792-5. The least weighted squares

15. Mašíček, L. (2003): Consistency of the least weighted squares estimator. To appear in Kybernetika. Plát, P. (2003): Nejmenší vážené čtverce. (The Least Weighted Squares, in Czech.) Diploma thesis on the Faculty of Nuclear and Physical Engineering, he Czech Technical University, Prague Mašíček,, L. (2003): Diagnostika a sensitivita robustního odhadu. (Diagnostics and sensitivity of robust estimators, in Czech) Dissertation on the Faculty of Mathematics, Charles University. The least weighted squares Both, in the framework of random carriers as well as for deterministc ones we have consistency, asymptotic normality and Bahadur representation of the Least weighted Squares. There are also some optimality results Mašíček,, L. (2003): Optimality of the least weighted squares estimator. To appear in the Proceedings of ICORS'2003.

16. The least weighted squares There is also algorithm for evaluating the LEAST WEIGHTED SQUARES. It is a modification of the algorithm for the LEAST TRIMMED SQUARES which was described and tested in: If interested in, ask me for sending a copy. Víšek, J.Á. (1996): On high breakdown point estimation. Computational Statistics (1996) 11:137-146. Víšek, J.Á. (2000): On the diversity of estimates. CSDA 34, (2000) 67 - 89. Čížek, P., J. Á. Víšek (2000): The least trimmed squares. User Guide of Explore, Humboldt University. (Of course, the algorithm for LTS is available in the package EXPLORE.)

17. The least weighted squares - algorithm Select randomly p + 1 observations and find regression plane through them. A Put Is this sum of weighted squared residuals smaller than the sum from the previous step? B Evaluate squared residuals for all observations, order these squared residuals from the largest one to the smallest, multiply them by the weights and evaluate the sum of these products. No Order observations in the same order as the squared residuals and apply the classical weighted least squares on them with weights and find new regression plane. Yes

18. Return to Continued Have we found already 20 identical models or have we exhausted a priori given number of repetitions ? The least weighted squares - algorithm End of evaluation YesNo A B The algorithm is available in MATLAB. In the case when we were able to pass all n! orders of observations ( less than 10 observations), i.e. when we were able to find the LEAST WEIGHTED SQUARES estimator precisely, the algorithm returned the same value. Continued An arbitrary reasonable number

19. Recalling Cragg’s idea Accommodating Cragg’s idea for robust regression

20. Recalling classical weighted least squares Accomodying Cragg’s idea for robust regression

21. The least weighted squares & Cragg’s idea The first step

22. The least weighted squares & Cragg’s idea The second step continued &

24. THANKS for ATTENTION