Abstract: In this paper, we study how to detect gradual changes in linear models. Firstly, the test statistics based on the weighted residual part and the weighted-CUSUM are proposed. Secondly, it is proved that the test statistics converge in distribution to the standard Brownian bridge under the original assumption, converge in distribution to the Brownian bridge with a drift term under the alternative hypothesis. The conditions for the test consistence are also given. Finally, simulation results show that the proposed method is feasible.

Key words: gradual change in structure, Weighted-CUSUM, linear models, Brownian bridge, consistency