Two Simple Approximations to the Distributions of Quadratic Forms

Many test statistics are asymptotically equivalent to quadratic forms of normal variables, which are further equivalent to T = ∑di=1 Λiz2i with zi 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 Λi's and the degrees of freedom d by analysis and Monte Carlo. The results imply that one approximation can be as good as the exact distribution when d is large. When the coefficient of variation of the Λi's is small, another approximation is also adequate for practical model inference. The results are applied to a study of alcoholism and psychological symptoms