IDEAS home Printed from https://ideas.repec.org/a/eee/csdana/v53y2009i4p853-856.html

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167-9473(08)00565-3
    Download Restriction: Full text for ScienceDirect subscribers only.
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    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. 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.
    5. Hui-Guo Zhang & Chang-Lin Mei, 2017. "Discussion," International Statistical Review, International Statistical Institute, vol. 85(1), pages 38-40, April.
    6. 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.
    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. 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.
    9. Wolfgang Kössler & Hans-J. Lenz & Xing D. Wang, 2024. "Some new invariant sum tests and MAD tests for the assessment of Benford’s law," Computational Statistics, Springer, vol. 39(7), pages 3779-3800, December.
    10. 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.
    11. 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).
    12. 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.
    13. Yamaguchi, Hikaru & Murakami, Hidetoshi, 2023. "The multi-aspect tests in the presence of ties," Computational Statistics & Data Analysis, Elsevier, vol. 180(C).
    14. Songhua Tan & Qianqian Zhu, 2022. "Asymmetric linear double autoregression," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(3), pages 371-388, May.
    15. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Shively, Philip A., 2000. "Stationary time-varying risk premia in forward foreign exchange rates," Journal of International Money and Finance, Elsevier, vol. 19(2), pages 273-288, April.
    2. Escanciano, Juan Carlos & Jacho-Chávez, David T., 2010. "Approximating the critical values of Cramér-von Mises tests in general parametric conditional specifications," Computational Statistics & Data Analysis, Elsevier, vol. 54(3), pages 625-636, March.
    3. Hong Zhang & Zheyang Wu, 2023. "The generalized Fisher's combination and accurate p‐value calculation under dependence," Biometrics, The International Biometric Society, vol. 79(2), pages 1159-1172, June.
    4. Chin-Shang Li, 2016. "A test for the linearity of the nonparametric part of a semiparametric logistic regression model," Journal of Applied Statistics, Taylor & Francis Journals, vol. 43(3), pages 461-475, March.
    5. Taamouti, Abderrahim, 2009. "Analytical Value-at-Risk and Expected Shortfall under regime-switching," Finance Research Letters, Elsevier, vol. 6(3), pages 138-151, September.
    6. Nicolas Städler & Sach Mukherjee, 2017. "Two-sample testing in high dimensions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(1), pages 225-246, January.
    7. Mahmood Kharrati-Kopaei, 2021. "On the exact distribution of the likelihood ratio test statistic for testing the homogeneity of the scale parameters of several inverse Gaussian distributions," Computational Statistics, Springer, vol. 36(2), pages 1123-1138, June.
    8. Dufour, Jean-Marie & Taamouti, Abderrahim, 2008. "Exact optimal and adaptive inference in regression models under heteroskedasticity and non-normality of unknown forms," UC3M Working papers. Economics we086027, Universidad Carlos III de Madrid. Departamento de Economía.
    9. Elodie Persyn & Richard Redon & Lise Bellanger & Christian Dina, 2018. "The impact of a fine-scale population stratification on rare variant association test results," PLOS ONE, Public Library of Science, vol. 13(12), pages 1-17, December.
    10. El-Bassiouni, M. Y. & Charif, H. A., 2004. "Testing a null variance ratio in mixed models with zero degrees of freedom for error," Computational Statistics & Data Analysis, Elsevier, vol. 46(4), pages 707-719, July.
    11. A. Basu & A. Mandal & N. Martin & L. Pardo, 2018. "Testing Composite Hypothesis Based on the Density Power Divergence," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(2), pages 222-262, November.
    12. Tamar Sofer & Elizabeth D. Schifano & David C. Christiani & Xihong Lin, 2017. "Weighted pseudolikelihood for SNP set analysis with multiple secondary outcomes in case‐control genetic association studies," Biometrics, The International Biometric Society, vol. 73(4), pages 1210-1220, December.
    13. Peng, Qidi & Schellhorn, Henry, 2018. "On the distribution of extended CIR model," Statistics & Probability Letters, Elsevier, vol. 142(C), pages 23-29.
    14. Shively, Philip A., 2002. "An exact invariant variance ratio test," Economics Letters, Elsevier, vol. 75(3), pages 347-353, May.
    15. Lu, Zeng-Hua, 2006. "The numerical evaluation of the probability density function of a quadratic form in normal variables," Computational Statistics & Data Analysis, Elsevier, vol. 51(3), pages 1986-1996, December.
    16. Abhijit Mandal, 2020. "An optimal test for the additive model with discrete or categorical predictors," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(6), pages 1397-1417, December.
    17. David Preinerstorfer, 2018. "How to avoid the zero-power trap in testing for correlation," Papers 1812.10752, arXiv.org.
    18. David E. A. Giles, 2006. "The Exact Asymptotic Distribution Function of Watson's UN-Squared for Testing Goodness-of-Fit With Circular Discrete Data," Econometrics Working Papers 0607, Department of Economics, University of Victoria.
    19. Francesca Torti & Domenico Perrotta & Marco Riani & Andrea Cerioli, 2019. "Assessing trimming methodologies for clustering linear regression data," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 13(1), pages 227-257, March.
    20. 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.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:csdana:v:53:y:2009:i:4:p:853-856. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/csda .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.