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

We propose a non-parametric local method for estimating change points on th e mean of a sequence of independent r.v. (X-l)(i=1),...,(n). Making the dif ference 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-f unction" 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 p oint 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. Sinc e the test and the change point estimator take only into account informatio n on boxes of size 2A these procedures are local and therefore well adapted to the case of more than one change point.