MCMC-CE: A Novel and Efficient Algorithm for Estimating Small Right-Tail Probabilities of Quadratic Forms with Applications in Genomics

ABSTRACT Quadratic forms of multivariate normal variables play a critical role in statistical applications, particularly in genomics and bioinformatics. However, accurately computing small right-tail probabilities ( p -values) for large-scale quadratic forms is computationally challenging due to the intractability of their probability distributions, as well as significant numerical constraints and computational burdens. To address these problems, we propose MCMC-CE, an innovative algorithm that integrates Markov Chain Monte Carlo (MCMC) sampling with the cross-entropy (CE) method, coupled with leading-eigenvalue extraction and Satterthwaite-type approximation techniques. Our approach efficiently estimates small p -values for quadratic forms with their ranks exceeding 10,000. Through extensive simulation studies and real-world applications in genomics, including genome-wide association studies and pathway enrichment analyses, our method demonstrates advantageous numerical accuracy and computational reliability compared to existing approaches such as Davies’, Imhof’s, Farebrother’s, Liu-Tang-Zhang’s and saddlepoint approximation methods. MCMC-CE provides a robust and scalable solution for accurately computing small p -values for quadratic forms, facilitating more precise statistical inference in large-scale genomic studies.