Optimal nonparametric change point analysis
Oscar Hernán Madrid Padilla, Yi Yu, Daren Wang, Alessandro Rinaldo
Abstract
We study change point detection and localization for univariate data in fully nonparametric settings, in which at each time point, we acquire an independent and identically distributed sample from an unknown distribution that is piecewise constant. The magnitude of the distributional changes at the change points is quantified using the Kolmogorov–Smirnov distance. Our framework allows all the relevant parameters, namely the minimal spacing between two consecutive change points, the minimal magnitude of the changes in the Kolmogorov–Smirnov distance, and the number of sample points collected at each time point, to change with the length of the time series. We propose a novel change point detection algorithm based on the Kolmogorov–Smirnov statistic and show that it is nearly minimax rate optimal. Our theory demonstrates a phase transition in the space of model parameters. The phase transition separates parameter combinations for which consistent localization is possible from the ones for which this task is statistically infeasible. We provide extensive numerical experiments to support our theory.