LAUSR

The conditional copula of a random pair $$(Y_1,Y_2)$$(Y1,Y2) given the value taken by some covariate $$X \in {\mathbb {R}}$$X∈R is the function $$C_x:[0,1]^2 \rightarrow [0,1]$$Cx:[0,1]2→[0,1] such that $${\mathbb {P}}(Y_1 \le y_1, Y_2 \le y_2 | X=x) = C_x \{ {\mathbb {P}}(Y_1\le y_1 | X=x), {\mathbb {P}}(Y_2\le y_2 | X=x) \}$$P(Y1≤y1,Y2≤y2|X=x)=Cx{P(Y1≤y1|X=x),P(Y2≤y2|X=x)}. In this note, the weak convergence of the two estimators of $$C_x$$Cx proposed by Gijbels et al. (Comput Stat Data Anal 55(5):1919–1932, 2011) is established under $$\alpha $$α-mixing. It is shown that under appropriate conditions on the weight functions and on the mixing coefficients, the limiting processes are the same as those obtained by Veraverbeke et al. (Scand J Stat 38(4):766–780, 2011) under the i.i.d. setting. The performance of these estimators in small sample sizes is investigated with simulations.