A local method for estimating change points: The "hat-function"

We propose a non-parametric local method for estimating change points on the mean of a sequence of independent r.v. (X-l)(i=1),...,(n). Making the difference between the moving average of X-l on a right box and a left box of size A we gel a function k bar right arrow D(A, k) which presents as "hat-function" at each change point. We estimate the change point as the maximum of D(A,k) after thresholding. We give bounds on the error for this change point estimator. As a by-product, we build a test of the existence of abrupt changes and we gel bounds on the error probabilities of type 1 and 2. Since the test and the change point estimator take only into account information on boxes of size 2A these procedures are local and therefore well adapted to the case of more than one change point.