Robins James, Li Lingling, Tchetgen Eric, van der Vaart Aad
Departments of Biostatistics and Epidemiology, School of Public Health, Harvard University, Mathematical Institute, Leiden University.
Stoch Process Their Appl. 2016 Dec;126(12):3733-3759. doi: 10.1016/j.spa.2016.04.005.
We prove conditional asymptotic normality of a class of quadratic U-statistics that are dominated by their degenerate second order part and have kernels that change with the number of observations. These statistics arise in the construction of estimators in high-dimensional semi- and non-parametric models, and in the construction of nonparametric confidence sets. This is illustrated by estimation of the integral of a square of a density or regression function, and estimation of the mean response with missing data. We show that estimators are asymptotically normal even in the case that the rate is slower than the square root of the observations.
我们证明了一类二次U统计量的条件渐近正态性，这类统计量由其退化的二阶部分主导，并且核随观测值数量而变化。这些统计量出现在高维半参数和非参数模型中估计量的构造，以及非参数置信集的构造中。通过对密度或回归函数平方的积分估计以及缺失数据下平均响应的估计来说明这一点。我们表明，即使在速率慢于观测值平方根的情况下，估计量也是渐近正态的。
