Dyadic data is often encountered when quantities of interest are associated with the edges of a network. As such it plays an important role in statistics, econometrics and many other data science disciplines. We consider the problem of uniformly estimating a dyadic Lebesgue density function, focusing on nonparametric kernel-based estimators taking the form of dyadic empirical processes. Our main contributions include the minimax-optimal uniform convergence rate of the dyadic kernel density estimator, along with strong approximation results for the associated standardized and Studentized $t$-processes. A consistent variance estimator enables the construction of valid and feasible uniform confidence bands for the unknown density function. A crucial feature of dyadic distributions is that they may be "degenerate" at certain points in the support of the data, a property making our analysis somewhat delicate. Nonetheless our methods for uniform inference remain robust to the potential presence of such points. For implementation purposes, we discuss procedures based on positive semi-definite covariance estimators, mean squared error optimal bandwidth selectors and robust bias-correction techniques. We illustrate the empirical finite-sample performance of our methods both in simulations and with real-world data. Our technical results concerning strong approximations and maximal inequalities are of potential independent interest.
翻译:当兴趣量与网络边缘相关时,往往会遇到Dyadi数据。因此,它在统计、计量经济学和许多其他数据科学学科中起着重要作用。我们认为统一估算dyadic Lebesgue密度函数的问题,重点是以dyadic 经验过程为形式的非对等内核测算器。我们的主要贡献包括dyadic内核密度估计器的最小最大和最佳统一趋同率,以及相关标准化和学生化的美元进程的强烈近似结果。一致的差异估计器有助于为未知密度函数构建有效和可行的统一信任带。dyadic Lebesgue分布的一个关键特征是,它们可能在某些支持数据的地方“降解”以内核为主,而这种属性使我们的分析有些微妙。尽管我们的统一推断方法对于这些点的潜在存在仍然十分有力。为了执行目的,我们讨论基于积极的半确定性估算器的近似近似近似近结果的程序,意味着为未知密度功能的未知的密度功能构建有效和可行的统一信任带宽度。 dyadicredi 分布的一个关键特征是,在数据支持数据的某些点上,我们的最佳选择器和稳健健的精确的精确度模拟方法是我们关于真实数据模拟方法。