# zbMATH — the first resource for mathematics

Large adaptive estimation in linear regression model. I: Consistency. (English) Zbl 0792.62033
Summary: A condition of identifiability of linear regression models with symmetric distribution of errors is given. Following R. Beran’s approach [Ann. Stat. 6, 292-313 (1978; Zbl 0378.62051)] for the location case, consistency and asymptotic normality of this adaptive estimator are proved. The result shows that the estimator is not asymptotically efficient. But it selects a model with such distribution function of errors which is (in the sense of Hellinger distance applied to $$F(x)$$ and $$1-P(-x))$$ “as much as possible symmetric”, which may be useful when we know that there are no reasons for the asymmetry.

##### MSC:
 62G07 Density estimation 62G20 Asymptotic properties of nonparametric inference 62G35 Nonparametric robustness 62F35 Robustness and adaptive procedures (parametric inference)
Full Text:
##### 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 $$L_{1}$$-norm based. Statistical Data Analysis Based on the $$L_{1}$$-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 Schätzer in Regression Modellen. Ph. D. Diseration, Institut für Math. Stochastik, Universität Hamburg 1988. [7] R. Koenker: A lecture read on 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: Limit theorems for stochastic processes. Teor. Veroyatnost. i Primenen. 1 (1956), 261 - 290. · Zbl 0074.33802 [14] C. Stein: Efficient nonparametric testing and estimation. Proc. Third Berkeley Symp. Math. Statist. 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 Automation, Czechoslovak 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. To appear in the Proceedings of the Second International Workshop on Model-Oriented Data Analysis, Springer-Verlag, Wien. · Zbl 0801.62058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.