Kendall’s tau-based inference for gradually changing dependence structures

Suppose that a sequence of random pairs $$(X_1,Y_1)$$ ( X 1 , Y 1 ) , $$\ldots $$ … , $$(X_n,Y_n)$$ ( X n , Y n ) is subject to a gradual change in the sense that for $$K_1 \le K_2 \in \{ 1, \ldots , n \}$$ K 1 ≤ K 2 ∈ { 1 , … , n } , the joint distribution is F before $$K_1$$ K 1 , G after $$K_2$$ K 2 , and gradually moving from F to G between the two times of change $$K_1$$ K 1 and $$K_2$$ K 2 . This setup elegantly generalizes the abrupt-change model that is usually assumed in the change-point analysis. Under this configuration, asymptotically unbiased estimates of Kendall’s tau up to and after the change are proposed, 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.