Uniform saddlepoint approximations for ratios of quadratic forms

TL;DR

This paper develops uniform saddlepoint approximations for the distribution of ratios of quadratic forms in correlated normal variables, ensuring accurate error bounds across the entire support range.

Contribution

It introduces saddlepoint approximations that maintain uniform accuracy for ratios of quadratic forms, including explicit error bounds at support edges.

Findings

Approximate c.d.f. and density with uniform relative error

Explicit limiting error values at support edges

Applicable to various serial correlation statistics

Abstract

Ratios of quadratic forms in correlated normal variables which introduce noncentrality into the quadratic forms are considered. The denominator is assumed to be positive (with probability 1). Various serial correlation estimates such as least-squares, Yule--Walker and Burg, as well as Durbin--Watson statistics, provide important examples of such ratios. The cumulative distribution function (c.d.f.) and density for such ratios admit saddlepoint approximations. These approximations are shown to preserve uniformity of relative error over the entire range of support. Furthermore, explicit values for the limiting relative errors at the extreme edges of support are derived.