Approximations to the distribution of a large quadratic form in normal variables

 

The 2018 Joint NZSA/ORSNZ Conference

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 \lambda_{1},...,\lambda_n of \Sigma A, where A is the matrix representing the quadratic form and \Sigma 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(n^3) to O(n^2k) on extracting eigenvalues and avoids speed problems with computing the sum of n terms. For accuracy, the existing methods have some limits on 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 \Sigma in the extreme right tail, so it is usable for arbitrarily small p-values.