Change-point detection for long-range dependent sequences in a general setting

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 nonstationary. We study a class of nonparametric estimators and prove that 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. © 2009 Elsevier Ltd. All rights reserved.