We delve into the estimation of the functional coefficients and inference for varying coefficient model. Applying Laguerre series, we develop an estimator for the vector of functional coefficients that attains asymptotically optimal convergence rates in the minimax sense. These rates are derived for functional coefficients that belong to Laguerre-Sobolev space. The method is based on approximating the functional coefficients using truncated Laguerre series and choosing empirical Laguerre coefficients that minimize the least squares criterion. In addition, we establish the asymptotic normality of the estimator for the functional coefficients, construct their confidence intervals, and establish point-wise hypothesis tests about their true values. A simulations study is carried out to examine the finite-sample properties of the proposed methodology. A real data set is considered as well, and results based on the proposed methodology are compared to those based on selected existing approaches.

翻译：本文深入研究了变系数模型中函数系数的估计与推断问题。通过应用拉盖尔级数展开，我们构建了一种函数系数向量的估计器，该估计器在最小最大意义下达到了渐近最优收敛速率。这些速率是针对属于拉盖尔-索伯列夫空间的函数系数推导得出的。该方法基于截断拉盖尔级数逼近函数系数，并通过最小化最小二乘准则来选择经验拉盖尔系数。此外，我们建立了函数系数估计器的渐近正态性，构建了其置信区间，并建立了关于其真实值的逐点假设检验。通过模拟研究检验了所提方法在有限样本下的性质。同时分析了一个实际数据集，并将基于所提方法得到的结果与选定的现有方法进行了比较。