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

TL;DR

This paper develops and compares various approximations for calculating boundary crossing probabilities of moving sums of i.i.d. normal variables, highlighting a new correction method that performs well even with small window sizes.

Contribution

The paper introduces a novel correction approximation for discrete boundary crossing probabilities based on continuous Gaussian process theory, improving accuracy in small window scenarios.

Findings

The new approximation outperforms existing methods in accuracy.

It remains highly accurate even for small window lengths.

Numerical results validate the effectiveness of the proposed approach.

Abstract

In this paper 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.