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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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