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From moments of sum to moments of product

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  • Kan, Raymond

Abstract

We provide an identity that relates the moment of a product of random variables to the moments of different linear combinations of the random variables. Applying this identity, we obtain new formulae for the expectation of the product of normally distributed random variables and the product of quadratic forms in normally distributed random variables. In addition, we generalize the formulae to the case of multivariate elliptically distributed random variables. Unlike existing formulae in the literature, our new formulae are extremely efficient for computational purposes.

Suggested Citation

  • Kan, Raymond, 2008. "From moments of sum to moments of product," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 542-554, March.
    • Handle: RePEc:eee:jmvana:v:99:y:2008:i:3:p:542-554
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    References listed on IDEAS

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    1. Schott, James R., 2003. "Kronecker product permutation matrices and their application to moment matrices of the normal distribution," Journal of Multivariate Analysis, Elsevier, vol. 87(1), pages 177-190, October.
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    3. Jan R. Magnus, 1979. "The expectation of products of quadratic forms in normal variables: the practice," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 33(3), pages 131-136, September.
    4. Jan R. Magnus, 1986. "The Exact Moments of a Ratio of Quadratic Forms in Normal Variables," Annals of Economics and Statistics, GENES, issue 4, pages 95-109.
    5. Berkane, Maia & Bentler, P. M., 1986. "Moments of elliptically distributed random variates," Statistics & Probability Letters, Elsevier, vol. 4(6), pages 333-335, October.
    6. repec:adr:anecst:y:1986:i:4:p:05 is not listed on IDEAS
    7. Blacher, René, 2003. "Multivariate quadratic forms of random vectors," Journal of Multivariate Analysis, Elsevier, vol. 87(1), pages 2-23, October.
    8. Jan R. Magnus, 1986. "The Exact Moments of a Ratio of Quadratic Forms in Normal Variables," Annals of Economics and Statistics, GENES, issue 4, pages 95-109.
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    Cited by:

    1. Baishuai Zuo & Chuancun Yin & Narayanaswamy Balakrishnan, 2020. "Explicit expressions for joint moments of $n$-dimensional elliptical distributions," Papers 2007.09349, arXiv.org, revised Aug 2020.
    2. Mutschler, Willi, 2015. "Note on Higher-Order Statistics for the Pruned-State-Space of nonlinear DSGE models," VfS Annual Conference 2015 (Muenster): Economic Development - Theory and Policy 113138, Verein für Socialpolitik / German Economic Association.
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    5. David Rossell & Oriol Abril & Anirban Bhattacharya, 2021. "Approximate Laplace approximations for scalable model selection," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(4), pages 853-879, September.
    6. Russell Oliver & Sun Wei, 2024. "Using sums-of-squares to prove Gaussian product inequalities," Dependence Modeling, De Gruyter, vol. 12(1), pages 1-13.
    7. Robert E. Gaunt, 2022. "The basic distributional theory for the product of zero mean correlated normal random variables," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 76(4), pages 450-470, November.
    8. Grant Hillier & Raymond Kan, 2021. "Moments of a Wishart Matrix," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 19(1), pages 141-162, December.
    9. Kan, Raymond & Wang, Xiaolu, 2010. "On the distribution of the sample autocorrelation coefficients," Journal of Econometrics, Elsevier, vol. 154(2), pages 101-121, February.
    10. Mutschler, Willi, 2015. "Identification of DSGE models—The effect of higher-order approximation and pruning," Journal of Economic Dynamics and Control, Elsevier, vol. 56(C), pages 34-54.
    11. Song, Iickho & Lee, Seungwon, 2015. "Explicit formulae for product moments of multivariate Gaussian random variables," Statistics & Probability Letters, Elsevier, vol. 100(C), pages 27-34.
    12. repec:nbp:nbpbik:v:47:y:2016:i:6:p:365-394 is not listed on IDEAS
    13. Grant Hillier & Raymond Kan & Xiaolu Wang, 2008. "Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors," CeMMAP working papers CWP14/08, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    14. Hillier, Grant & Kan, Raymond & Wang, Xiaoulu, 2009. "Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors," Discussion Paper Series In Economics And Econometrics 918, Economics Division, School of Social Sciences, University of Southampton.
    15. Vignat, C., 2012. "A generalized Isserlis theorem for location mixtures of Gaussian random vectors," Statistics & Probability Letters, Elsevier, vol. 82(1), pages 67-71.
    16. Hillier, Grant & Kan, Raymond & Wang, Xiaoulu, 2009. "Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors," Discussion Paper Series In Economics And Econometrics 0918, Economics Division, School of Social Sciences, University of Southampton.
    17. Christian Gische & Manuel C. Voelkle, 2022. "Beyond the Mean: A Flexible Framework for Studying Causal Effects Using Linear Models," Psychometrika, Springer;The Psychometric Society, vol. 87(3), pages 868-901, September.
    18. Oh Kang Kwon & Stephen Satchell, 2020. "The Distribution of Cross Sectional Momentum Returns When Underlying Asset Returns Are Student’s t Distributed," JRFM, MDPI, vol. 13(2), pages 1-19, February.
    19. Łukasz Lenart & Agnieszka Leszczyńska-Paczesna, 2016. "Do market prices improve the accuracy of inflation forecasting in Poland? A disaggregated approach," Bank i Kredyt, Narodowy Bank Polski, vol. 47(5), pages 365-394.
    20. Seth Pruitt, 2012. "Uncertainty Over Models and Data: The Rise and Fall of American Inflation," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 44, pages 341-365, March.
    21. Lucio Fernandez‐Arjona & Damir Filipović, 2022. "A machine learning approach to portfolio pricing and risk management for high‐dimensional problems," Mathematical Finance, Wiley Blackwell, vol. 32(4), pages 982-1019, October.
    22. Jiayu Lai & Xiaoyi Wang & Kaige Zhao & Shurong Zheng, 2023. "Block-diagonal test for high-dimensional covariance matrices," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(1), pages 447-466, March.
    23. Edelmann, Dominic & Richards, Donald & Royen, Thomas, 2023. "Product inequalities for multivariate Gaussian, gamma, and positively upper orthant dependent distributions," Statistics & Probability Letters, Elsevier, vol. 197(C).
    24. Julia Adamska & Łukasz Bielak & Joanna Janczura & Agnieszka Wyłomańska, 2022. "From Multi- to Univariate: A Product Random Variable with an Application to Electricity Market Transactions: Pareto and Student’s t -Distribution Case," Mathematics, MDPI, vol. 10(18), pages 1-29, September.

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