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Nonparametric Estimates of Regression Quantiles and Their Local Bahadur Representation

Probal Chaudhuri
The Annals of Statistics
Vol. 19, No. 2 (Jun., 1991), pp. 760-777
Stable URL: http://www.jstor.org/stable/2242082
Page Count: 18
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Nonparametric Estimates of Regression Quantiles and Their Local Bahadur Representation
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Abstract

Let (X, Y) be a random vector such that X is d-dimensional, Y is real valued and Y = θ(X) + ε, where X and ε are independent and the αth quantile of ε is 0 (α is fixed such that $0 < \alpha < 1$). Assume that θ is a smooth function with order of smoothness $p > 0$, and set r = (p - m)/(2p + d), where m is a nonnegative integer smaller than p. Let T(θ) denote a derivative of θ of order m. It is proved that there exists a pointwise estimate T̂n of T(θ), based on a set of i.i.d. observations (X1, Y1),⋯,(Sn, Yn), that achieves the optimal nonparametric rate of convergence n-r under appropriate regularity conditions. Further, a local Bahadur type representation is shown to hold for the estimate T̂n and this is used to obtain some useful asymptotic results.

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