α-Ball divergence and its applications to change-point problems for Banach-valued sequences
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In this paper, we extend a measure of divergence between two distributions: Ball divergence, to a new one: α-Ball divergence. With this new notion, we propose its sample statistic which can be used to test whether two weakly dependent sequences of Banach-valued random vectors have the same distribution. The properties of α-Ball divergence and its sample statistic, as Ball divergence has, are inspected and shown to hold for random sequences which are functionals of some absolutely regular sequences. We further apply the sample statistic to change-point problems for a sequence of weakly dependent Banach-valued observations with multiple possible change-points. Our procedure does not require any assumptions on special change-point type. It could detect the number of change-points as well as their locations. We also prove the consistency of the estimated change-point locations. Extensive simulation studies and analyses of two interesting real data sets about wind direction and bitcoin price illustrate that our procedure has considerable advantages over other existing competitors, especially when observations are non-Euclidean or there are distributional changes in the variance.
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