LINEAR ESTIMATORS IN CHANGE-POINT PROBLEMS

Observations X(i) are uncorrelated with means theta(i), i = 1,...,n, a nd variances 1. The linear estimators theta = TX, for some n x n matri x T, are widely used in smoothing problems, where it is assumed that n eighbouring parameter Values are similar The smoothness assumption is violated in change point problems, where neighbouring parameter values are equal, except at some unspecified change points where there are j umps of unknown size from one parameter value to the next. In the case of a single change point in one dimension, for any linear estimator, the expected sum of squared errors between estimates and parameters is of order root n for some choice of parameters, compared to order 1 fo r the least squares estimate. We show similar results for adaptive shi ft estimators, in which the linear estimator uses a kernel estimated f rom the data. Finally, for a change point problem in two dimensions, t he expected sum of squared errors is of order n3/4.