Abstract

Suppose that a sequence of random pairs (X1, Y1), . . ., (Xn, Yn) is subject to a gradual change in the sense that for K1 ≤ K2 ∈ {1,...,n}, the joint distribution of a pair is F before K1, G after K2, and gradually moving from F to G between the two times of change K1 and K2. This setup elegantly generalizes the abrupt-change model that is usually assumed in the change- point analysis. This configuration gives asymptotically unbiased estimates of Kendall’s tau up to the change and after the change, as well as tests and estimators of change points related to these measures. The asymptotic behaviour of the introduced estimators and test statistics is rigorously investigated, in particular by demonstrating a general result on weighted indexed U-statistics computed under a heterogeneous pattern. A simulation study is conducted to examine the sampling properties of the proposed methods under different scenarios of change in the dependence structure of bivariate series. An illustration is given on a time series of monthly atmospheric carbon dioxide concentrations and global temperature for the period 1959–2015.