Concentration of Randomized Functions of Uniformly Bounded Variation
with Thomas Anton, Sutanuka Roy
Abstract: A sharp, distribution-free, non-asymptotic result is proved for the concentration of a random function around the mean function when the randomization is generated by a finite sequence of independent data and the random functions satisfy uniform bounded variation assumptions. This work is primarily motivated by the need for inference on the distributional impacts of social policy interventions. However, the family of randomized functions that we study is broad enough to cover wide-ranging applications. For example, we provide a Kolmogorov–Smirnov-like test for continuous-time samples and Lipschitz-continuous stochastic processes, alongside novel tools for inference on the evolution of treatment effects over time. A Dvoretzky–Kiefer– Wolfowitz like inequality is also provided for the sum of almost surely monotone random functions, extending the famous non-asymptotic work of Massart (1990) for empirical cumulative distribution functions generated by i.i.d.data, to, inter alia, settings without micro-clusters proposed by Canay, Santos, and Shaikh (2018). We illustrate the relevance of our theoretical results for applied work via empirical applications.