On the connection between orthant probabilities and the first passage time problem

TL;DR

This paper introduces a Monte Carlo method that estimates orthant probabilities by simulating Gaussian time-series crossing times, offering a potentially faster alternative to existing methods, demonstrated on ARFIMA models.

Contribution

The paper presents a novel Monte Carlo approach linking orthant probabilities to first passage times in Gaussian time-series, improving computational speed.

Findings

Method outperforms existing software in speed

Uses Davies-Harte algorithm for ARFIMA models

Accurately estimates orthant probabilities via simulation

Abstract

This article describes a new Monte Carlo method for the evaluation of the orthant probabilities by sampling first passage times of a non-singular Gaussian discrete time-series across an absorbing boundary. This procedure makes use of a simulation of several time-series sample paths, aiming to record their first crossing instants. Thus, the computation of the orthant probabilities is traced back to the accurate simulation of a non-singular Gaussian discrete-time series. Moreover, if the simulation is also efficient, this method is shown to be more speedy than the others proposed in the literature. As example, we make use of the Davies-Harte algorithm in the evaluation of the orthant probabilities associated to the ARFIMA$(0,d,0)$ model. Test results are presented that compare this method with currently available software.