摘要：This paper provides an accessible methodology for approximating the distribution ofa general linear combination of non-central chi-square random variables. Attentionis focused on the main application of the results, namely the distribution of p ositivedefinite and indefinite quadratic forms in normal random variables. After explainingthat the moments of a quadratic form can b e determined from its cumulants by meansof a recursive formula, we propose a moment-based approximation of the density func-tion of a positive definite quadratic form, which consists of a gamma density functionthat is adjusted by a linear combination of Laguerre polynomials or, equivalently, bya single polynomial. On expressing an indefinite quadratic form as the di.erence oftwo positive definite quadratic forms, explicit representations of approximations to itsdensity and distribution functions are obtained in terms of con.uent hypergeometricfunctions. The proposed closed form expressions converge rapidly and provide accu-rate approximations over the entire support of the distribution. Additionally, b oundsare derived for the integrated squared and absolute truncation errors. An easilyimplementable algorithm is provided and several illustrative numerical examples arepresented. In particular, the methodology is applied to the Durbin–Watson statistic.Finally, relevant computational considerations are discussed. Linear combinations ofchi-square random variables and quadratic forms in normal variables being ubiquitousin statistics, the distribution approximation technique introduced herewith shouldprove widely applicable.
关键词：chi-square random variables; linear combinations; quadratic forms; cumulants;moments; density approximation; Durbin–Watson statistic
