  Detecting the emergence of an abrupt change-point is a classic problem in
statistics and machine learning. Kernel-based nonparametric statistics have
been used for this task which enjoy fewer assumptions on the distributions than
the parametric approach and can handle high-dimensional data. In this paper we
focus on the scenario when the amount of background data is large, and propose
two related computationally efficient kernel-based statistics for change-point
detection, which are inspired by the recently developed $B$-statistics. A novel
theoretical result of the paper is the characterization of the tail probability
of these statistics using the change-of-measure technique, which focuses on
characterizing the tail of the detection statistics rather than obtaining its
asymptotic distribution under the null distribution. Such approximations are
crucial to control the false alarm rate, which corresponds to the significance
level in offline change-point detection and the average-run-length in online
change-point detection. Our approximations are shown to be highly accurate.
Thus, they provide a convenient way to find detection thresholds for both
offline and online cases without the need to resort to the more expensive
simulations or bootstrapping. We show that our methods perform well on both
synthetic data and real data.
