Chen, Tong

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

Tong Chen and Thomas Lumley

Department of Statistics, University of Auckland

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 \(\lambda_{1},\ldots,\lambda_n\) of \(\Sigma A\), where \(A\) is the matrix representing the quadratic form and \(\Sigma\) is the covariance matrix of the Gaussians. There is quite a bit of literature on this problem, but it mostly deals with approximations on small quadratic forms \((n < 10)\) and moderate p-values \((p > 10^{-2})\). Motivated by genetic applications, we look at large quadratic forms \((n > 1000)\) and very small p-values \((p < 10^{-4})\). We compare existing methods under genetic settings and show that a leading-eigenvalue approximation, which only takes the largest \(k\) eigenvalues, has computational advantage without any 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 approximating the sum of \(n\) terms. For accuracy, under large quadratic forms, moment methods are inaccurate for very small p-values and Farebrother’s method is not useable if the minimum eigenvalue is small, but the saddlepoint approximation has bounded relative error even in the extreme right tail.

This presentation is eligible for the NZSA Student Prize.