部分函数线性模型的模型平均方法

Optimal Model Averaging Estimation for Partial Functional Linear Models

部分函数线性模型是一种被广泛研究和应用的模型，其响应变量与一般的随机变量有关，也与函数型的随机变量有关．文章首先利用传统的谱分解方法来表示协方差函数，将部分函数线性模型的函数部分线性化，其次基于Hansen（2007）的Mallows模型平均方法，提出了该模型下的最优权重的选择准则，并证明了模型平均估计量的渐近最优性，此外还考虑了候选模型为两个特殊模型的情况下的模型平均估计量的渐近最优性．最后，进行了模拟研究，并对肉类和玉米样本的近红外反射光谱数据集进行分析，均表明所提出的模型平均方法是有效的．

In this paper, we propose a model averaging method for the partial functional linear models （PFLM）, which are designed for the case that the scalar response is related to a vector of random variables and some function-valued random variables as predictor variables. First, we use a conventional spectral decomposition for the covariance functions, and linearize the PFLM with inner product of the functional variables and the eigenfunctions. The least squares estimators are obtained by the candidate linearized models. Then, we propose an optimal weight choice criterion for the PFLM, which is based on the Mallows＇ criterion proposed by Hansen （2007） and derive the asymptotic optimality of the model average estimators. Besides, we also prove the asymptotic optimality of the model average estimators under the case which only two special candidate models are considered. The proposed method is illustrated with a simulation study, and is applied to the data set of near infrared reflection （NIR） spectra of meat samples and corn samples.