Approximations for the boundary crossing probabilities of moving sums of normal random variables

Abstract

In this article we study approximations for boundary crossing probabilities for the moving sums of i.i.d. normal random variables. We propose approximating a discrete time problem with a continuous time problem allowing us to apply developed theory for stationary Gaussian processes and to consider a number of approximations (some well known and some not). We bring particular attention to the strong performance of a newly developed approximation that corrects the use of continuous time results in a discrete time setting. Results of extensive numerical comparisons are reported. These results show that the developed approximation is very accurate even for small window length.

Keywords:

• Moving sum
• Boundary crossing probability
• Moving sum of normal
• Change-point detection

Mathematics Subject Classification (2000):

• Primary: 60G50
• 60G35
• Secondary: 60G70
• 94C12
• 93E20

Acknowledgments

The authors are grateful to our colleague Nikolai Leonenko for intelligent discussions and finding the reference (Harrison 1985), which is essential for the material of Section 4.2.