Distribution of a Sum of Weighted Chi-Square Variables

Abstract

Abstract We consider distributions of quadratic forms of the type Qk = Σ k j = 1 cj (xj + aj )2, where the xj 's are independent and identically distributed standard normal variables, and where cj and aj are nonnegative constants. Exact significance points of Qk , for selected values of cj and all aj = 0, have been published for k = 2, 3, 4, and 5. We give significance points for k = 6, 8, and 10. We propose and assess two new approximations to Qk : (1) fitting a Pearson curve with the same first four moments as Qk ; and (2) fitting Qk = Awr , where w has the χ2 p distribution, and where A, r, and p are determined by the first three moments of Qk.