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Adaptive estimation in linear regression model. II: Asymptotic normality. (English) Zbl 0792.62034
Summary: [For part I see the preceding review, Zbl 0792.62033.]
An asymptotic representation of an adaptive estimator based on R. Beran’s idea [Ann. Stat. 6, 292-313 (1978; Zbl 0378.62051)] of minimizing Hellinger distance is derived. It is shown that the estimator is asymptotically normal but not efficient. From the practical point of view the approach may be useful because it selects a model with distribution of residuals symmetric “as much as possible” (in the sense of Hellinger distance applied to \(F(x)\) and \(1-F(x)\)). It is not difficult to construct a numerical example showing that sometimes it is the only way how to find a proper model.

MSC:
62G07 Density estimation
62G35 Nonparametric robustness
62G20 Asymptotic properties of nonparametric inference
62F35 Robustness and adaptive procedures (parametric inference)
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References:
[1] R. Beran: An efficient and robust adaptive estimator of location. Ann. Statist. 6 (1978), 292 - 313. · Zbl 0378.62051
[2] P. J. Bickel: The 1980 Wald Memorial Lectures - On adaptive estimation. Ann. Statist. 10 (1982), 647-671. · Zbl 0489.62033
[3] Y. Dodge: An introduction to statistical data analysis ii-norm based. Statistical Data Analysis Based on the Ii-norm and Related Methods (Y. Dodge, North-Holland, Amsterdam 1987. · Zbl 0615.00009
[4] J. Jurečková: Regression quantiles and trimmed least square estimator under a general design. Kybernetika 20 (1984), 345 - 357. · Zbl 0561.62027
[5] F. R. Hampel E. M. Ronchetti P. J. Rousseeuw, W. A. Stahel: Robust Statistics: The Approach Based on Influence Functions. J. Wiley, New York 1986. · Zbl 0593.62027
[6] G. Heimann: Adaptive und robuste Schatzer in Regression modellen. Ph.D. Dissertation, Institut fur Math. Stochastik, Universitat Hamburg 1988.
[7] R. Koenker: A lecture read at Charles University during his visit in Prague 1989. · Zbl 0736.62060
[8] R. Koenker, G. Basset: Regression quantiles. Econometrica 46 (1978), 33 - 50. · Zbl 0373.62038
[9] H. Koul, F. DeWet: Minimum distance estimation in a linear regression model. Ann. Statist. 11 (1983), 921 -932. · Zbl 0521.62023
[10] R. A. Maronna, V. J. Yohai: Asymptotic behaviour of general M-estimates for regression and scale with random carriers. Z. Wahrsch. verw. Gebiete 58 (1981), 7 - 20. · Zbl 0451.62031
[11] P.J. Rousseeuw, A.M. Leroy: Robust Regression and Outlier Detection. J. Wiley, New York 1987. · Zbl 0711.62030
[12] D. Ruppert, R. J. Carroll: Trimmed least squares estimation in linear model. J. Amer. Statist. Assoc. 75 (1980), 828-838. · Zbl 0459.62055
[13] A. V. Skorokhod: Limìt theoremsfor stochastic processes. Teor. Veroyatnost. i Primenen. 1 (1956), 261 - 290.
[14] C. Stein: Efficient nonparametric testing and estimation. Proc. Third Berkeley Symp. Math. Statìst. Prob. 1 (1956), 187 - 196. Univ. of California Press, Berkeley, Calif. 1956. · Zbl 0074.34801
[15] C. Stone: Adaptive maximum likelihood estimators of a location parameter. Ann. Statist. 3 (1975), 267 - 284. · Zbl 0303.62026
[16] J. Á. Víšek: What is adaptivity of regression analysis intended for?. Transactions of ROBUST’90, JČMF, Prague 1990, pp. 160 - 181.
[17] J. Á. Víšek: Adaptive Estimation in Linear Regression Model. Research Report No. 1642, Institute of Information Theory and Automatiou, Czechosbvak Academy of Sciences, Prague 1990.
[18] J. Á. Víšek: Adaptive Maximum-likelihood-like Estimation in Linear Model. Research Report No. 1654, Institute of Information Theory and Automation, Czechoslovak Academy of Sciences, Prague 1990.
[19] J. Á. Víšek: Adaptive estimation in linear regression model and test of symmetry of residuals. Proceedings of the Second International Workshop on Model-Oriented Data Anaylsis, Saint Kyrik, Plovdiv, Bulgaria 1990
[20] J. Á. Víšek: Adaptive estimation in linear regression model. Part 1: Consistency. Kybernetika 28 (1992), 1,26-36. · Zbl 0792.62033
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