Simultaneous estimation of multiple conditional regression quantiles (Q1987595)

The authors consider the simultaneous estimation of multiple non-parametric quantile functions in the model \[ y_i = \mu \left(x_i\right) + \epsilon_i, \qquad 1 \leq i \leq n, \] where the error contributions \(\epsilon_i\) are assumed to be i.i.d. If, for a fixed \(\tau \in \left(0,1\right)\), the function \(\mu \left(x,\tau\right)\) denotes the corresponding quantile curve, the considered estimator is based on minimizing \[ \int_0^1 \sum_{j=1}^n \rho_\tau \left(y_i - \mu\left(x_i, \tau\right)\right) d\tau \] with the loss \(\rho_\tau \left(u\right)=u\left(\tau-1_{u \leq 0}\right)\). The minimization of this functional is performed by fitting tensor product bivariate B-splines to \(\mu\) and then adding a roughness penalty to obtain smoothness of the estimator. The corresponding smoothing parameters are chosen via a Schwartz information criterion. The authors derive the asymptotic behavior of the proposed estimator and demonstrate its performance on a real data set.