Computing the distribution of quadratic forms: Further comparisons between the Liu–Tang–Zhang approximation and exact methods

Liu, Tang and Zhang [Liu, H. Tang, Y., Zhang H.H. 2009. A new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables. Computational Statistics & Data Analysis 53, 853–856] proposed a chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables. To approximate the distribution of interest, they used a non-central chi-square distribution, where the degrees of freedom and the non-centrality parameter were calculated using the first four cumulants of the quadratic form. Numerical examples were encouraging, suggesting that the approximation was particularly accurate in the upper tail of the distribution. We present here additional empirical evidence, comparing Liu–Tang–Zhang’s four-moment non-central chi-square approximation with exact methods. While the moment-based method is interesting because of its simplicity, we demonstrate that it should be used with care in practical work, since numerical examples suggest that significant differences may occur between that method and exact methods, even in the upper tail of the distribution.