Inference for Change-Point and Post-Change Mean with Possible Change in Variance

Abstract

For a sequence of independent normal random variables, we consider the estimation of the change-point and the post-change mean after a change in the mean is detected by a CUSUM procedure, subject to a possible change in variance. Conditional on the event that a change is detected and it occurred far away from the starting point and the threshold is large, the (absolute) bias of the maximum likelihood estimator for the change-point (obtained at the reference value) is found. The first-order biases for the post-change mean and variance estimators are also obtained by using Wald's Likelihood Ratio Identity and the renewal theorem. In the local case when the reference value and the post-change mean are both small, accurate approximations are derived. A confidence interval for post-change mean based on a corrected normal pivot is then discussed.

Keywords:

• Biased estimation
• Change-point problem
• Corrected confidence interval
• CUSUM procedure
• Random walk theory
• Renewal theorem

Subject Classification:

• 62L12

ACKNOWLEDGMENTS

The author is grateful to three referees for their careful reading, and to the editor for many editorial helps. The main results of the paper form a part of some lecture notes (Wu, 2005) written by the author.