Zhu Liping, Huang Mian, Li Runze
School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai, 200433, P. R. China.
Department of Statistics and The Methodology Center, The Pennsylvania State University, University Park, Pennsylvania, 16802-2111, USA.
Stat Sin. 2012 Oct;22(4):1379-1401. doi: 10.5705/ss.2010.199.
This paper is concerned with quantile regression for a semiparametric regression model, in which both the conditional mean and conditional variance function of the response given the covariates admit a single-index structure. This semiparametric regression model enables us to reduce the dimension of the covariates and simultaneously retains the flexibility of nonparametric regression. Under mild conditions, we show that the simple linear quantile regression offers a consistent estimate of the index parameter vector. This is a surprising and interesting result because the single-index model is possibly misspecified under the linear quantile regression. With a root- consistent estimate of the index vector, one may employ a local polynomial regression technique to estimate the conditional quantile function. This procedure is computationally efficient, which is very appealing in high-dimensional data analysis. We show that the resulting estimator of the quantile function performs asymptotically as efficiently as if the true value of the index vector were known. The methodologies are demonstrated through comprehensive simulation studies and an application to a real dataset.
本文关注半参数回归模型的分位数回归，其中给定协变量时响应变量的条件均值和条件方差函数都具有单指标结构。这种半参数回归模型使我们能够降低协变量的维度，同时保留非参数回归的灵活性。在温和条件下，我们表明简单线性分位数回归为指标参数向量提供了一致估计。这是一个令人惊讶且有趣的结果，因为在线性分位数回归下，单指标模型可能设定错误。有了指标向量的根一致估计，就可以采用局部多项式回归技术来估计条件分位数函数。该过程计算效率高，在高维数据分析中非常有吸引力。我们表明，由此得到的分位数函数估计量在渐近意义上与指标向量真实值已知时一样有效。通过全面的模拟研究和对真实数据集的应用展示了这些方法。
