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Convergence rate of \(B\)-spline estimators of nonparametric conditional quantile functions. (English) Zbl 1383.62111
Summary: Given a bivariate sample \(\{(X_i,Y_i),i=1,2,\dots,n\}\), we consider the problem of estimating the conditional quantile functions of nonparametric regression by minimizing \(\sum\rho_\alpha(Y_i-g(X_i))\) over \(g\) in a linear space of B-spline functions, where \(\rho_\alpha(u)=| u|-(2\alpha-1)u\) is the Czech function of R. Koenker and G. Bassett jun. [Econometrica 46, 33–50 (1978; Zbl 0373.62038)]. If the true conditional quantile function is smooth up to order \(r\), we show that the optimal global convergence rate of \(n^{-r/(2r+1)}\) is attained by the B-spline based estimators if the number of knots is in the order of \(n^{1/(2r+1)}\).

MSC:
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
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