Detection of change-points near the end points of long-range dependent sequences

We consider a sequence of observations (X_i)_{i = 1, ..., n} with a marginal distribution that is given by L (X_i) = P_n if i ≤ n θ_n and L (X_i) = Q_n if i > n θ_n. The parameter 0 <θ_n <1 is the location of the change-point which must be estimated and may depend on the sequence length. We consider the general case in which the change-point can converge to one of the end-points of the interval [0, 1] as the sequence length n tends to infinity. The sequence can be long-range dependent, short-range dependent or independent and may be non-stationary. We study a class of non-parametric estimators and prove they are consistent and that the rate of convergence is 1 / n. We also deal with the case in which the distance between the distributions P_n and Q_n tends to zero as n tends to infinity. To cite this article: W. Nie et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009). © 2009 Académie des sciences.