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An iteration method of data censoring in the regression estimation problem. (English. Russian original) Zbl 1127.62037
Autom. Remote Control 68, No. 4, 645-656 (2007); translation from Avtom. Telemekh. 68, No. 4, 79-91 (2007).
Summary: A robust analog of the Nadaraya-Watson regression estimate [E. A. Nadaraya, Theor. Probab. Appl. 10, 186–190 (1965; Zbl 0134.36302); G. S. Watson, Sankhyā, Ser. A 26, 359–372 (1964; Zbl 0137.13002)] is considered. A solution is obtained in the class of censoring algorithms. A criterion and iteration procedure for determining a censored sample are proposed. The criterion is based on the analysis of the residuals (errors) of estimation.
MSC:
62G08 Nonparametric regression and quantile regression
65C60 Computational problems in statistics (MSC2010)
62G35 Nonparametric robustness
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[1] Hampel, F., Ronchetti, E., Rousseeuw, P., and Stahel, W., Robust Statistics. The Approach on Influence Function, New York: Willey, 1986. · Zbl 0593.62027
[2] Huber, P., Robust Statistics, New York: Willey, 1981. · Zbl 0536.62025
[3] Tukey, J.W., The Future of the Data Analysis, Ann. Math. Stat., 1962, vol. 33(1), pp. 1–67. · Zbl 0107.36401
[4] Box, G.E.P., Non-normality and Tests on Variances, Biometrics, 1953, vol. 40, pp. 318–335. · Zbl 0051.10805
[5] Kirik, E.S., On Nonparametrical Approach to Modeling of a Robust Estimate of Regression, Trudy 3 mezhdunarodnoi konferentsii ”Identifikatsiya sistem i zadachi upravleniya–SICPRO’04” (Proc. 3rd Int. Conf. ”Identificiation of Systems and Control Problems–SICPRO’04”), Moscow: Inst. Probl. Upravlen., 2004, pp. 811–844.
[6] Stone, C.J., Consistent Nonparametric Regression, Ann. Statist., 1977, vol. 5(4), pp. 595–645. · Zbl 0366.62051
[7] Stone, C. J., Optimal Global Rates of Convergence for Nonparametric Regression, Ann. Statist., 1982, vol. 10, pp. 1040–1053. · Zbl 0511.62048
[8] Nadaraya, E.A., On Nonparametric Estimates of Probability Densitity and Regression, Teor. Veroyatn. Primenen., 1965, vol. 10(1), pp. 199–203.
[9] Watson, G., Smooth Regression Analysis, Sankhya, Ser. A, 1965, vol. 26, no. 4, pp. 359–372. · Zbl 0137.13002
[10] Ruban, A.I., Identification of Stochastic Objects on the Basis of the Nonparametric Approach, Avtom. Telemekh., 1979, no. 11, pp. 106–118.
[11] Epanechnikov, V.A., Nonparametric Estimation of a Multidimensional Probability Density, Teor. Veroyatn. Primenen., 1968, vol. 14, pp. 156–161. · Zbl 0175.17101
[12] Hardle, W. and Marron, J.S., Asymptotic Nonequivalents of Some Bandwidth Selection, Biometrica, 1985, vol. 72, pp. 481–484. · Zbl 0571.62034
[13] Hardle, W. and Marron, J.S., Optimal Bandwidth Selection in Nonparametric Regression Function Estimation, Ann. Statist., 1985, vol. 13, pp. 1465–1481. · Zbl 0594.62043
[14] Kirik, E.S., On Nonparametrical Approach to the Robust Regression Estimating, Proc. IASTED Int. Conf. ”Automation, Control, and Information Technology,” Anaheim, 2002, pp. 294–299.
[15] Medvedev, A.V., Neparametricheskie sistemy adaptatsii (Nonparametric Adaptation Systems), Novosibirsk: Nauka, 1983. · Zbl 0526.93002
[16] Andrews, D.F., A Robust Method for Multiple Linear Regression, Technometrics, 1974, vol. 16(4), pp. 523–531. · Zbl 0294.62082
[17] Cleveland, W.S., Robust Locally Weighted Regression and Smoothing Scatterplots, J. Am. Statist. Associat., 1979, vol. 74, pp. 829–836. · Zbl 0423.62029
[18] Hinich, M.J. and Talvar, P.P., A Simple Method for Robust Regression, J. Am. Statist. Associat., 1975, vol. 70(379), pp. 113–119. · Zbl 0309.62045
[19] Hogg, R.V., Adaptive Robust Procedure: A Partial Review and Some Suggestions for Future Applications and Theory, J. Am. Statist. Associat., 1974, vol. 69, pp. 909–923. · Zbl 0305.62030
[20] Jaeckel, L.A., Some Flexible Estimates of Location, Ann. Math. Stat., 1971, vol. 42, pp. 1540–1552. · Zbl 0232.62008
[21] Rousseuw, P.J. and Zomeren, B.C., Unmasking Multivariate Outliers and Leverage Points, J. Am. Statist. Associat., 1990, vol. 85(411), pp. 633–639.
[22] Yale, C. and Forsythe, A.B., Winsorized Regression, Technometrics, 1976, vol. 18(3), pp. 291–300. · Zbl 0342.62044
[23] Lepskii, O.V., Mammen, E., and Spokoiny, V.G., Ideal Spatial Adaptation to Inhomogeneous Smothness: An Approach Based on Kernal Estimates with Variable Bandwidth Selection, Ann. Math. Stat., 1997, vol. 25(3), pp. 929–947. · Zbl 0885.62044
[24] Lepskii, O.V. and Spokoiny, V.G., Optimal Pointwise Adaptive Methods in Nonparametric Estimation, Ann. Math. Stat., 1997, vol. 25(6), pp. 2512–2546. · Zbl 0894.62041
[25] Katkovnik, V.Ya., Neparametricheskaya identifikatsiya i sglazhivanie dannykh (Nonparametric Identification and Data Smoothing), Moscow: Nauka, 1985. · Zbl 0576.62050
[26] Tsybakov, A.B., On Convergence of Nonparametric Robust Algorithms of Reconstruction of Functions, Avtom. Telemekh., 1983, no. 9, pp. 3–46. · Zbl 0539.62047
[27] Hardle, W., Applied Nonparametric Regression, New York: Cambridge Univ. Press, 1990.
[28] Anscombe, F.J., Rejection of Outliers, Technometrics, 1960, vol. 2(2), pp. 159–166. · Zbl 0091.14806
[29] Bickel, P.J., On Some Robust Estimates of Location, Ann. Math. Stat., 1965, vol. 36, pp. 847–858. · Zbl 0192.25802
[30] Grubbs, F.E., Procedure for Detecting Outlying Observations in Samples, Technometrics, 1969, vol. 11(1), pp. 1–21.
[31] McMillan, R.G., Test for One or Two Outliers in Normal Samples with Unknown Variance, Technometrics, 1971, vol. 49, pp. 87–100. · Zbl 0232.62005
[32] Rosner, B.R., On Detection of Many Outliers, Technometrics, 1975, vol. 17(2), pp. 221–227. · Zbl 0308.62025
[33] Tietjen, G.L. and Moore, R.H., Some Grubbs-Type Statistics for the Detections of Several Outliers, Technometrics, 1972, vol. 55, pp. 583–598.
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