Two simple approximations to the distributions of quadratic forms

Abstract

Many test statistics are asymptotically equivalent to quadratic forms of normal variables, which are further equivalent to with z_i being independent and following N(0,1). Two approximations to the distribution of T have been implemented in popular software and are widely used in evaluating various models. It is important to know how accurate these approximations are when compared to each other and to the exact distribution of T. The paper systematically studies the quality of the two approximations and examines the effect of the λ_i and the degrees of freedom d by analysis and Monte Carlo. The results imply that the adjusted distribution for T can be as good as knowing its exact distribution. When the coefficient of variation of the λ_i is small, the rescaled statistic is also adequate for practical model inference. But comparing T_R against will inflate type I errors when substantial differences exist among the λ_i, especially, when d is also large.