ESTIMATION IN FUNCTIONAL LINEAR QUANTILE REGRESSION
ESTIMATION IN FUNCTIONAL LINEAR QUANTILE REGRESSION
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DOI：
10.1214/12-aos1066
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发表时间：
2012-12-01
影响因子：
4.5
通讯作者：
Kato, Kengo
中科院分区：
文献类型：
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作者：
Kato, Kengo
本文研究因变量为标量，协变量为函数的函数线性分位数回归中的估计问题，将每个固定分位数指标的条件分位数建模为协变量的线性泛函。在这里，我们假设协变量是离散观察的，采样点可能在不同的受试者中不同，其中每个受试者的测量数量随着样本量的增加而增加。此外，我们允许分位数指数在开放单位区间的给定子集上变化，因此斜率函数是两个变量的函数：(通常)时间和分位数指数。同样，条件分位数函数是分位数指数和协变量的函数。我们考虑基于主分量基的斜率函数的一个估计量。通过插入法获得条件分位数函数的估计器。由于所构造的插件估计不一定满足关于分位数指标的单调性约束，我们还考虑了一类条件分位数函数的单调化估计。我们在适当的范数下建立了这些估计量的收敛速度，证明了这些估计在协变量的协方差核和斜率函数的某些光滑性假设下在极小极大意义下是最优的。用模拟的方法研究了截止水平的经验选择。
This paper studies estimation in functional linear quantile regression in which the dependent variable is scalar while the covariate is a function, and the conditional quantile for each fixed quantile index is modeled as a linear functional of the covariate. Here we suppose that covariates are discretely observed and sampling points may differ across subjects, where the number of measurements per subject increases as the sample size. Also, we allow the quantile index to vary over a given subset of the open unit interval, so the slope function is a function of two variables: (typically) time and quantile index. Likewise, the conditional quantile function is a function of the quantile index and the covariate. We consider an estimator for the slope function based on the principal component basis. An estimator for the conditional quantile function is obtained by a plug-in method. Since the so-constructed plug-in estimator not necessarily satisfies the monotonicity constraint with respect to the quantile index, we also consider a class of monotonized estimators for the conditional quantile function. We establish rates of convergence for these estimators under suitable norms, showing that these rates are optimal in a minimax sense under some smoothness assumptions on the covariance kernel of the covariate and the slope function. Empirical choice of the cutoff level is studied by using simulations.