Detecting a smooth signal: Optimality of cusum based procedures

Abstract

Cox et al. (1988) show that under a random smooth function model with signal covariance matrix V , a locally most powerful test for the presence of a nonpolynomial signal is based on the statistic y T V y. We show that if a well-known difference-based roughness measure is used to derive a natural choice for V then this statistic is the sum of squares of a residual cusum sequence. Thus we confirm theoretically the long-held impression that the residual cusum sequence is powerful as a tool for the detection of smooth signal.