Statistica Sinica 2(1992), 69-90

ASYMPTOTIC APPROXIMATIONS FOR LIKELIHOOD

RATIO TESTS AND CONFIDENCE REGIONS FOR
A CHANGE-POINT IN THE MEAN OF
A MULTIVARIATE NORMAL DISTRIBUTION

Barry James, Kang Ling James and David Siegmund

University of Minnesota, University of Minnesota and
Stanford University

Abstract: For independent, d-dimensional normally distributed observations we give a tail approximation for the significance level of the likelihood ratio test of no change in the mean vector against the alternative of exactly one change. Assuming there is exactly one change in the mean vector, we obtain conditional likelihood ratio confidence regions for the change-point and joint regions for the change-point and size of the change. For the significance level we compare our approximation numerically with the improved Bonferroni upper bound of Srivastava and Worsley (1986). For our probability calculations we adapt the method of Woodroofe (1976, 1978).

Key words and phrases: Change-point, boundary crossing probability, likelihood ratio test.