dc.contributor.author |
Chen, Tong |
en |
dc.contributor.author |
Lumley, Thomas |
en |
dc.date.accessioned |
2019-10-01T10:59:11Z |
en |
dc.date.issued |
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 |
http://hdl.handle.net/2292/48222 |
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 |
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm |
en |
dc.rights.uri |
https://creativecommons.org/licenses/by-nc-nd/4.0/ |
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 |
http://purl.org/eprint/accessRights/OpenAccess |
en |
pubs.subtype |
Article |
en |
pubs.elements-id |
772019 |
en |
pubs.org-id |
Science |
en |
pubs.org-id |
Statistics |
en |
dc.identifier.eissn |
1872-7352 |
en |
pubs.record-created-at-source-date |
2019-05-13 |
en |
pubs.online-publication-date |
2019-05-11 |
en |