Chen, TongLumley, Thomas2019-10-012019-11Computational Statistics and Data Analysis 139:75-81 Nov 20190167-9473https://hdl.handle.net/2292/48222Quadratic forms of Gaussian variables occur in a wide range of applications in statistics. They can be expressed as a linear combination of chi-squareds. The coefficients in the linear combination are the eigenvalues λ1,…,λn of ΣA , where A is the matrix representing the quadratic form and Σ is the covariance matrix of the Gaussians. The previous literature mostly deals with approximations for small quadratic forms (n<10) and moderate p-values (p>10−2) . Motivated by genetic applications, moderate to large quadratic forms ( 300<n<12,000 ) and small to very small p-values (p<10−4) are studied. Existing methods are compared under these settings and a leading-eigenvalue approximation, which only takes the largest k eigenvalues, is shown to have the computational advantage without any important loss in accuracy. For time complexity, a leading-eigenvalue approximation reduces the computational complexity from O(n3) to O(n2k) on extracting eigenvalues and avoids speed problems with computing the sum of n terms. For accuracy, the existing methods have some limits in calculating small p-values under large quadratic forms. Moment methods are inaccurate for very small p-values, and Farebrother’s method is not usable if the minimum eigenvalue is much smaller than others. Davies’s method is usable for p-values down to machine epsilon. The saddlepoint approximation is proved to have bounded relative error for any A and Σ in the extreme right tail, so it is usable for arbitrarily small p-values.Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated. Previously published items are made available in accordance with the copyright policy of the publisher.https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmhttps://creativecommons.org/licenses/by-nc-nd/4.0/Journal Article10.1016/j.csda.2019.05.002Copyright: Elsevierhttp://purl.org/eprint/accessRights/OpenAccess1872-7352