Although semiparametric models, in particular varying-coefficient models, alleviate the curse of dimensionality by avoiding estimation of fully nonparametric multivariate functions, there would typically still be a large number of functions to estimate. We propose a dimension reduction approach to estimating a large number of nonparametric univariate functions in varying-coefficient models, in which these functions are constrained to lie in a finite-dimensional subspace consisting of the linear span of a small number of smooth functions. The proposed methodology is put in the context of quantile regression, which provides more information on the response variable than the more conventional mean regression. Finally, we present some numerical illustrations to demonstrate the performances.

尽管半参数模型，特别是变系数模型，通过避免估计完全非参数的多元函数来缓解维数灾难，但通常仍然需要估计大量函数。我们提出了一种降维方法来估计变系数模型中的大量非参数单变量函数，其中这些函数被约束在有限维子空间中，该子空间由少量平滑函数的线性跨度组成。所提出的方法被置于分位数回归的背景下，与更传统的平均回归相比，它提供了更多关于响应变量的信息。最后，我们提供了一些数值插图来演示性能。