Method of Moments and Statistical Computations

Gene Golub (Stanford University)

Many statistical computations arising in linear least squares require the estimation of a quadratic form. We consider the problem of determining upper and lower bounds of the quadratic form u^{T} F(A) u where u is a given vector, A is a symmetric positive definite matix and F(.), an analytic function. The estimate of the quadratic form is determined by using Gauss quadrature rules; a basic tool of our studies is the Lanczos algorithm.

The technique we describe is useful in estimating the Generalized Cross Validation (GCV) function and solving linear least squares with a quadratic constraint. Our procedure is particular useful for large data sets.