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A new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables

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  • Liu, Huan
  • Tang, Yongqiang
  • Zhang, Hao Helen

Abstract

This note proposes a new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables. The unknown parameters are determined by the first four cumulants of the quadratic forms. The proposed method is compared with Pearson's three-moment central [chi]2 approximation approach, by means of numerical examples. Our method yields a better approximation to the distribution of the non-central quadratic forms than Pearson's method, particularly in the upper tail of the quadratic form, the tail most often needed in practical work.

Suggested Citation

  • Liu, Huan & Tang, Yongqiang & Zhang, Hao Helen, 2009. "A new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 853-856, February.
    • Handle: RePEc:eee:csdana:v:53:y:2009:i:4:p:853-856
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    References listed on IDEAS

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    1. Robert B. Davies, 1980. "The Distribution of a Linear Combination of χ2 Random Variables," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 29(3), pages 323-333, November.
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    4. Zhidong Bai & Guangming Pan & Yanqing Yin, 2018. "A central limit theorem for sums of functions of residuals in a high-dimensional regression model with an application to variance homoscedasticity test," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(4), pages 896-920, December.
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    6. Chen, Tong & Lumley, Thomas, 2019. "Numerical evaluation of methods approximating the distribution of a large quadratic form in normal variables," Computational Statistics & Data Analysis, Elsevier, vol. 139(C), pages 75-81.
    7. Sanae Rujivan & Athinan Sutchada & Kittisak Chumpong & Napat Rujeerapaiboon, 2023. "Analytically Computing the Moments of a Conic Combination of Independent Noncentral Chi-Square Random Variables and Its Application for the Extended Cox–Ingersoll–Ross Process with Time-Varying Dimens," Mathematics, MDPI, vol. 11(5), pages 1-29, March.
    8. Duchesne, Pierre & Lafaye De Micheaux, Pierre, 2010. "Computing the distribution of quadratic forms: Further comparisons between the Liu-Tang-Zhang approximation and exact methods," Computational Statistics & Data Analysis, Elsevier, vol. 54(4), pages 858-862, April.
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