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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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    1. 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.
    2. Zimmer, Zachary & Park, DoHwan & Mathew, Thomas, 2016. "Tolerance limits under normal mixtures: Application to the evaluation of nuclear power plant safety and to the assessment of circular error probable," Computational Statistics & Data Analysis, Elsevier, vol. 103(C), pages 304-315.
    3. Chenhua Li & Gary M. Gaukler & Yu Ding, 2013. "Using container inspection history to improve interdiction logistics for illicit nuclear materials," Naval Research Logistics (NRL), John Wiley & Sons, vol. 60(6), pages 433-448, September.
    4. Songhua Tan & Qianqian Zhu, 2022. "Asymmetric linear double autoregression," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(3), pages 371-388, May.
    5. Qianchuan He & Yang Liu & Ulrike Peters & Li Hsu, 2018. "Multivariate association analysis with somatic mutation data," Biometrics, The International Biometric Society, vol. 74(1), pages 176-184, March.
    6. Hui-Guo Zhang & Chang-Lin Mei, 2017. "Discussion," International Statistical Review, International Statistical Institute, vol. 85(1), pages 38-40, April.
    7. Solberger, M. & Zhou, X., 2013. "LM-type tests for idiosyncratic and common unit roots in the exact factor model with AR(1) dynamics," Research Memorandum 059, Maastricht University, Graduate School of Business and Economics (GSBE).
    8. 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.
    9. Wei Dai & Ming Yang & Chaolong Wang & Tianxi Cai, 2017. "Sequence robust association test for familial data," Biometrics, The International Biometric Society, vol. 73(3), pages 876-884, September.
    10. Xuemei Hu & Xiaohui Liu, 2013. "Empirical likelihood confidence regions for semi-varying coefficient models with linear process errors," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 25(1), pages 161-180, March.
    11. 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.
    12. 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.
    13. Huang, Peng & Gu, Yingkui & Li, He & Yazdi, Mohammad & Qiu, Guangqi, 2023. "An Optimal Tolerance Design Approach of Robot Manipulators for Positioning Accuracy Reliability," Reliability Engineering and System Safety, Elsevier, vol. 237(C).
    14. Yamaguchi, Hikaru & Murakami, Hidetoshi, 2023. "The multi-aspect tests in the presence of ties," Computational Statistics & Data Analysis, Elsevier, vol. 180(C).

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