Distribution of a Sum of Weighted Chi-Square Variables

Abstract

We consider distributions of quadratic forms of the type Q_k = Σ ^k _j = 1 c_j (x_j + a_j )², where the x_j 's are independent and identically distributed standard normal variables, and where c_j and a_j are nonnegative constants. Exact significance points of Q_k , for selected values of c_j and all a_j = 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 Q_k : (1) fitting a Pearson curve with the same first four moments as Q_k ; and (2) fitting Q_k = Aw^r , where w has the χ² _p distribution, and where A, r, and p are determined by the first three moments of Q_k.

Key Words:

• Quadratic forms
• Sums of weighted chi squares
• Imhof technqiue
• Pearson curves