title: Asymptotic Normality of Quadratic Estimators.
abstract: 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.
authors.affiliation: Departments of Biostatistics and Epidemiology, School of Public Health, Harvard University, Mathematical Institute, Leiden University.
authors.forename: James
authors.initials: J
authors.lastname: Robins
authors.affiliation: Departments of Biostatistics and Epidemiology, School of Public Health, Harvard University, Mathematical Institute, Leiden University.
authors.forename: Lingling
authors.initials: L
authors.lastname: Li
authors.affiliation: Departments of Biostatistics and Epidemiology, School of Public Health, Harvard University, Mathematical Institute, Leiden University.
authors.forename: Eric
authors.initials: E
authors.lastname: Tchetgen
authors.affiliation: Departments of Biostatistics and Epidemiology, School of Public Health, Harvard University, Mathematical Institute, Leiden University.
authors.forename: Aad
authors.initials: A
authors.lastname: van der Vaart
keywords: Projection estimator; Quadratic functional; Rate of convergence; U-statistic