PROPERTIES OF THE MINIMUM POINT OF AN UNBALANCED TWO-SIDED RANDOM WALK

Abstract

In this paper, properties of minimum point of a unbalanced two-sided random walk are investigated. Under the condition that the parameters at both sides tend to zero at the same order, probabilities that the minimum point is on which side, and the second order expansions for the first two moments of the minimum point are obtained. Applications of these results are very promising. First, they can be used to study the properties of the maximum likelihood estimator for the change point in the large sample case; second, they can be used to study inference problems after CUSUM test.

Keywords:

• Standard one parameter exponential family
• Ladder height and ladder epoch
• Strong renewal theorem
• CUSUM test

ACKNOWLEDGMENTS

This is part of my Ph.D. dissertation written at Univ. of Alberta under the supervision of Professor Yanhong Wu. I would like to express my sincere thanks for his encouragement and guidance.