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Optimal robust influence functions in semiparametric regression
Journal of Statistical Planning and Inference, vol. 140, no. 1, pp. 226-245
Robust statistics allows the distribution of the observations to be any member of a suitable neighborhood about an ideal model distribution. In this paper, the ideal models are semiparametric with finite-dimensional parameter of interest and a possibly infinite-dimensional nuisance parameter. In the asymptotic setup of shrinking neighborhoods, we derive and study the Hampel-type problem and the minmax MSE-problem. We show that, for all common types of neighborhood systems, the optimal influence function View the MathML sourceψ˜ can be approximated by the optimal influence functions View the MathML sourceψ˜n for certain parametric models. For general semiparametric regression models, we determine View the MathML source(ψ˜n)n∈N in case of error-in-variables and in case of error-free-variables. Finally, the results are applied to Cox regression where we compare our approach to that of Bednarski [1993. Robust estimation in Cox's regression model. Scand. J. Statist. 20, 213–225] in a small simulation study and on a real data set.