×

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 \(n^{-1/(2+d)}\) both pointwise and in the \(L^ q\) \((1\leq q<\infty)\) norm restricted to a compact. It can also be chosen to achieve the optimal rate of convergence \((n^{- 1}\log n)^{1/(2+d)}\) in the \(L^{\infty}\) norm restricted to a compact. These results also constitute an answer to an open question of C. J. Stone [ibid. 10, 1040-1053 (1982; Zbl 0511.62048)].

MSC:

62G05 Nonparametric estimation
62E20 Asymptotic distribution theory in statistics

Citations:

Zbl 0511.62048
PDFBibTeX XMLCite
Full Text: DOI