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The asymptotics for studentized \(k\)-step \(M\)-estimators of location. (English) Zbl 0839.62075

Summary: \(k\)-step versions \(M_n^{(k)}\) of studentized \(M\)-estimators \(M_n\) of location generated by various types of score functions are studied. The rate of approximation of \(M_n\) by \(M_n^{(k)}\) \((n \to \infty\), \(k\) fixed) is derived with the aid of the uniform asymptotic linearity results of three-parameters empirical \(M\)-processes. While the rate is considerably improved with increasing \(k\) for absolutely continuous score functions, it is improved very slowly for functions with jump-discontinuities where it can never be better than \(O_p (n^{- 1})\).

MSC:

62L12 Sequential estimation
62F12 Asymptotic properties of parametric estimators
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