Inference for post-change mean by a CUSUM procedure

In this paper, the inference problem for the post-change mean is considered after a change is detected by a CUSUM process in a sequence of independent normal variables. The change-point is estimated as the maximum likelihood estimate at the reference value and the post-change mean is estimated as the sample mean after the change-point estimate. By assuming the change-point is large and the monitoring limit approaches infinity, the first-order bias of the post-change mean estimate and a corrected asymptotic normal pivot are derived conditioning on that a change is detected. Local approximations for small reference value and post-change mean are obtained for numerical evaluation.

[1]  E. S. Page.  CONTINUOUS INSPECTION SCHEMES , 1954 .

[2]  M. Woodroofe,et al.  Corrected confidence intervals after sequential testing with applications to survival analysis , 1996 .

[3]  D. Siegmund.  Estimation following sequential tests , 1978 .

[4]  D. Siegmund.  Sequential Analysis: Tests and Confidence Intervals , 1985 .

[5]  Michael Woodroofe,et al.  Estimation after sequential testing: A simple approach for a truncated sequential probability ratio test , 1992 .

[6]  M. Woodroofe.  On stopping times and stochastic monotonicity , 1990 .

[7]  D. Hinkley.  Inference about the change-point from cumulative sum tests , 1971 .

[8]  Second Order Expansions for the Moments of Minimum Point of an Unbalanced Two-sided Normal Random Walk , 1999 .

[9]  Moshe Pollak,et al.  Convergence of quasi-stationary to stationary distributions for stochastically monotone Markov processes , 1986 .

[10]  Muni S. Srivastava,et al.  Quasi-stationary biases of change point and change magnitude estimation after sequential cusum test , 1999 .

[11]  John Whitehead,et al.  On the bias of maximum likelihood estimation following a sequential test , 1986 .

[12]  Michael Woodroofe,et al.  Corrected confidence sets for sequentially designed experiments , 1997 .

[13]  M. Woodroofe,et al.  Approximate bias calculations for sequentially designed experiments , 1998 .

[14]  John Whitehead,et al.  Confidence intervals for secondary parameters following a sequential test , 2000 .

[15]  Yanhong Wu.  Bias of estimator of change point detected by a CUSUM procedure , 2004 .