[Paper Review] Approximations of the boundary crossing probabilities for the maximum of moving sums

Jack Noonan, Anatoly Zhigljavsky|arXiv (Cornell University)|Oct 22, 2018

Hydrology and Drought Analysis1 citations

TL;DR

This paper proposes a novel approximation for boundary crossing probabilities of moving sums of i.i.d. normal random variables by bridging discrete-time processes with continuous-time theory for stationary Gaussian processes. It introduces a correction method that significantly improves accuracy in discrete settings, demonstrating high precision even for small window lengths through extensive numerical validation.

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.

Motivation & Objective

• To address the challenge of computing boundary crossing probabilities for discrete-time moving sums of i.i.d. normal random variables.
• To improve accuracy in discrete-time settings by adapting continuous-time results from stationary Gaussian process theory.
• To develop and evaluate a new correction technique that enhances the performance of continuous-time approximations in discrete contexts.
• To provide a numerically robust and efficient method for boundary crossing probability estimation, especially for small window lengths.

Proposed method

• The discrete-time moving sum process is approximated by a continuous-time Gaussian process to leverage established theoretical results.
• A correction term is introduced to adjust continuous-time approximations for the inherent discreteness of the original problem.
• Theoretical approximations are derived using properties of stationary Gaussian processes and their first-passage times.
• The method applies asymptotic and numerical techniques to evaluate and compare the accuracy of various approximations.
• Extensive numerical comparisons are conducted across different window lengths to assess performance.

Experimental results

Research questions

• RQ1 How accurately can continuous-time approximations estimate boundary crossing probabilities in discrete-time moving sum processes?
• RQ2 What is the impact of discretization error on standard continuous-time approximations in this context?
• RQ3 Can a correction term significantly improve the accuracy of continuous-time approximations when applied to discrete-time problems?
• RQ4 How does the performance of the proposed approximation vary with window length, especially for small samples?

Key findings

• The proposed correction method significantly enhances the accuracy of continuous-time approximations in discrete-time settings.
• The new approximation achieves high precision even for small window lengths, where standard methods often fail.
• Numerical comparisons show the proposed method outperforms both uncorrected continuous-time approximations and other known methods.
• The correction term effectively mitigates the bias introduced by the discretization of time in boundary crossing probability estimation.