Linear Estimators in Change Point Problems

Abstract

Observations $X_i$ are uncorrelated with means $\theta_i, i = 1,\cdots, n$, and variances 1. The linear estimators $\widehat{\theta} = T\mathbf{X}$, for some $n \times n$ matrix $T$, are widely used in smoothing problems, where it is assumed that neighbouring 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 jumps 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 $\sqrt n$ for some choice of parameters, compared to order 1 for the least squares estimate. We show similar results for adaptive shift estimators, in which the linear estimator uses a kernel estimated from the data. Finally, for a change point problem in two dimensions, the expected sum of squared errors is of order $n^{3/4}$.