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Expectations of products of quadratic forms in normal variables. (English) Zbl 0848.60019

Summary: Let \({\mathbf x}\) be normally distributed \(N_p (\mu, \Sigma)\) where \(\Sigma\) is positive definite and define the quadratic form \(Q ({\mathbf x}) = {\mathbf x}^T A{\mathbf x}\) and product \(Q_k = \prod^k_{i = 1} {\mathbf x}^T A_i {\mathbf x}\) of quadratic forms. We present explicit simple expressions for the raw moments and cumulants, of arbitrary order, of a quadratic form \(Q ({\mathbf x})\) in normal variates on \(R^p\) and of the expectation of a product \(Q_k\) of an arbitrary number of quadratic forms in normally distributed variables.

MSC:

60E99 Distribution theory
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[1] Don F.J.H., Statistica Neerlandica 33 pp 73– (1979) · Zbl 0416.62041
[2] Graybill F.A., Matrices with Applications in Statistics (1983) · Zbl 0496.15002
[3] Holmquist B., Linear and Multilinear Algebra 17 pp 117– (1985) · Zbl 0566.15012
[4] Holmquist, B. 1985.Moments and cumulants from generating functions of Hilbert space-valued random variables and an application to the Wishart distribution, Vol. 3, 1–86. University of Lund. Statistical Research Report
[5] Holmquist B., Stochastic Analysis and Applications 6 pp 273– (1988) · Zbl 0661.62036
[6] Jinadasa K.G., Stochastic Analysis and Applications 4 pp 399– (1986) · Zbl 0615.60014
[7] Johnson N.L., Distributions in Statistics:Continuous Univariate Distributions-2 (1970)
[8] Kathri C.G., Handbook of Statistics 1 pp 443– (1980)
[9] Kendall M.G., The Advanced Theory of Statistics 1 (1977)
[10] Li G., Inference in Elliptically Contoured and Related Distributions pp 433– (1990)
[11] Magnus J.R., Statistica Neerlandica 32 pp 201– (1978) · Zbl 0406.62031
[12] Magnus J.R., Statistica Neerlandica 33 pp 131– (1979) · Zbl 0412.60024
[13] Magnus J.R., Annals of Statistics 7 pp 381– (1979) · Zbl 0414.62040
[14] Mathai A.M., Quadratic forms in randorn variables (1992)
[15] Nagar A.L., Econometrika 27 pp 575– (1959) · Zbl 0091.15202
[16] Neudecker H., Statistica Neerlandica 22 pp 69– (1968) · Zbl 0153.49403
[17] Ruben H., Annals of Mathematical Statistics 33 pp 542– (1962) · Zbl 0117.37201
[18] Smith M.D., Forelaesninger over Almindlig Iaktaqelselaere:Sandsyldighetsregning ogmindste Kvadraters Methode (1889)
[19] Soong T.T., Stochastic Analysis and Applications 2 pp 295– (1984) · Zbl 0577.60041
[20] Thiele T.N., Forelaeaninger over Almindlig laktagelselaere: Sandsyldighetsregning og mindste Kvadraters Methode, (1889)
[21] Tracy D.S., Stochastic Analaysis and Applications 4 pp 111– (1986) · Zbl 0585.60024
[22] Tracy D.S., Stochastic Analysis and Applications 11 pp 337– (1993) · Zbl 0777.60017
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