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Asymptotic properties of kernel estimators based on local medians. (English) Zbl 0675.62031
Summary: The desire to make nonparametric regression robust leads to the problem of conditional median function estimation. Under appropriate regularity conditions, a sequence of local median estimators can be chosen to achieve the optimal rate of convergence $n\sp{-1/(2+d)}$ both pointwise and in the $L\sp q$ $(1\le q<\infty)$ norm restricted to a compact. It can also be chosen to achieve the optimal rate of convergence $(n\sp{- 1}\log n)\sp{1/(2+d)}$ in the $L\sp{\infty}$ norm restricted to a compact. These results also constitute an answer to an open question of {\it C. J. Stone} [ibid. 10, 1040-1053 (1982; Zbl 0511.62048)].

62G05Nonparametric estimation
62E20Asymptotic distribution theory in statistics
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