This paper proposes a moment-based approximation to the distribution of quadratic forms in gamma random variables. Quadratic forms in order statistics from an exponential population are considered as well. Actually, several test statistics can be expressed in terms of the latter. The density approximants are expressible as the product of a gamma type distributed base density function and a polynomial adjustment. Several illustrative examples are provided.
General representations of quadratic forms and quadratic expressions in singular normal vectors are given in terms of the difference of two positive definite quadratic forms and an independentlydistributed linear combination of normal random variables. Up to now, only special cases have been treated in the statistical literature. The densities of the quadratic forms are then approximated with gamma and generalized gamma density functions. A moment-based technique whereby the initial approximations are adjusted by means of polynomials is presented. Closed form and integral formulae are provided for the approximate density functions of the quadratic forms and quadratic expressions. A detailed step-by-step algorithm for implementing the proposed density approximation technique is also provided. Two numerical examples illustrate the methodology.
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