Numerical evaluation of methods approximating the distribution of a large quadratic form in normal variables

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Show simple item record Chen, Tong en Lumley, Thomas en 2019-10-01T10:59:11Z en 2019-11 en
dc.identifier.citation Computational Statistics and Data Analysis 139:75-81 Nov 2019 en
dc.identifier.issn 0167-9473 en
dc.identifier.uri en
dc.description.abstract Quadratic 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. en
dc.publisher Elsevier en
dc.relation.ispartofseries Computational Statistics and Data Analysis en
dc.rights 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. en
dc.rights.uri en
dc.rights.uri en
dc.title Numerical evaluation of methods approximating the distribution of a large quadratic form in normal variables en
dc.type Journal Article en
dc.identifier.doi 10.1016/j.csda.2019.05.002 en
pubs.begin-page 75 en
pubs.volume 139 en
dc.rights.holder Copyright: Elsevier en
pubs.end-page 81 en
dc.rights.accessrights en
pubs.subtype Article en
pubs.elements-id 772019 en Science en Statistics en
dc.identifier.eissn 1872-7352 en
pubs.record-created-at-source-date 2019-05-13 en 2019-05-11 en

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