On the biases of change point and change magnitude estimation after CUSUM test

The present paper investigates the biases of estimates of change point and change magnitude after CUSUM test. By assuming that the change point is far from the beginning and the in-control average run length of samples is large, second order approximations for the biases of both estimates are obtained by conditioning on detection, and biases of both estimates are very significant. Simulation studies show the approximations to be quite accurate in the case of detecting an increase in mean or variance when sampling from a normal distribution. The results demonstrate the fundamental differences between fixed sample size test and sequential test.