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

We consider a sequence of observations (Xi)i=1…n(Xi)i=1…n with a marginal distribution that is given by ℒ(Xi)=Pnℒ(Xi)=Pn if i≤nθni≤nθn and ℒ(Xi)=Qnℒ(Xi)=Qn if i>nθni>nθn. The parameter 0<θn<10<θn<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][0,1] as the sequence length nn 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/n1/n. We also deal with the case in which the distance between the distributions PnPn and QnQn tends to zero as nn tends to infinity.