IDEAS home Printed from https://ideas.repec.org/a/eee/csdana/v174y2022ics0167947322000172.html
   My bibliography  Save this article

Computing the asymptotic distribution of second-order U- and V-statistics

Author

Listed:
  • Seri, Raffaello

Abstract

Under general conditions, the asymptotic distribution of degenerate second-order U- and V-statistics is an (infinite) weighted sum of χ2 random variables whose weights are the eigenvalues of an integral operator associated with the kernel of the statistic. Also the behavior of the statistic in terms of power can be characterized through the eigenvalues and the eigenfunctions of the same integral operator. No general algorithm seems to be available to compute these quantities starting from the kernel of the statistic. An algorithm is proposed to approximate (as precisely as needed) the asymptotic distribution and to build several measures of performance for tests based on U- and V-statistics. The algorithm uses the Wielandt–Nyström method of approximation of an integral operator based on quadrature, and can be used with several methods of numerical integration. An extensive numerical study shows that the Wielandt–Nyström method based on Clenshaw–Curtis quadrature performs very well both for the eigenvalues and the eigenfunctions.

Suggested Citation

  • Seri, Raffaello, 2022. "Computing the asymptotic distribution of second-order U- and V-statistics," Computational Statistics & Data Analysis, Elsevier, vol. 174(C).
    • Handle: RePEc:eee:csdana:v:174:y:2022:i:c:s0167947322000172
    DOI: 10.1016/j.csda.2022.107437
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167947322000172
    Download Restriction: Full text for ScienceDirect subscribers only.
    File URL: https://libkey.io/10.1016/j.csda.2022.107437?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Colombi, Roberto, 2020. "Selection tests for possibly misspecified hierarchical multinomial marginal models," Econometrics and Statistics, Elsevier, vol. 16(C), pages 136-147.
    2. 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.
    3. Vuong, Quang H, 1989. "Likelihood Ratio Tests for Model Selection and Non-nested Hypotheses," Econometrica, Econometric Society, vol. 57(2), pages 307-333, March.
    4. Bruce Lindsay & Ramani Pilla & Prasanta Basak, 2000. "Moment-Based Approximations of Distributions Using Mixtures: Theory and Applications," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(2), pages 215-230, June.
    5. Dewan, Isha & Prakasa Rao, B. L. S., 2002. "Central limit theorem for U-statistics of associated random variables," Statistics & Probability Letters, Elsevier, vol. 57(1), pages 9-15, March.
    6. 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.
    7. Chen, Xiaohong & White, Halbert, 1998. "Central Limit And Functional Central Limit Theorems For Hilbert-Valued Dependent Heterogeneous Arrays With Applications," Econometric Theory, Cambridge University Press, vol. 14(2), pages 260-284, April.
    Full references (including those not matched with items on IDEAS)

    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. 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.
    2. Pötscher, Benedikt M. & Preinerstorfer, David, 2021. "Valid Heteroskedasticity Robust Testing," MPRA Paper 107420, University Library of Munich, Germany.
    3. 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.
    4. Roberto Colombi & Sabrina Giordano, 2019. "Likelihood-based tests for a class of misspecified finite mixture models for ordinal categorical data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(4), pages 1175-1202, December.
    5. Colombi, Roberto, 2020. "Selection tests for possibly misspecified hierarchical multinomial marginal models," Econometrics and Statistics, Elsevier, vol. 16(C), pages 136-147.
    6. Marco Riani & Andrea Cerioli & Domenico Perrotta & Francesca Torti, 2015. "Simulating mixtures of multivariate data with fixed cluster overlap in FSDA library," 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. 9(4), pages 461-481, December.
    7. Anton Rask Lundborg & Rajen D. Shah & Jonas Peters, 2022. "Conditional independence testing in Hilbert spaces with applications to functional data analysis," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(5), pages 1821-1850, November.
    8. Peng, Qidi & Schellhorn, Henry, 2018. "On the distribution of extended CIR model," Statistics & Probability Letters, Elsevier, vol. 142(C), pages 23-29.
    9. 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.
    10. Emanuele Taufer & Sudip Bose & Aldo Tagliani, 2009. "Optimal predictive densities and fractional moments," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 25(1), pages 57-71, January.
    11. Mnatsakanov, Robert M., 2008. "Hausdorff moment problem: Reconstruction of distributions," Statistics & Probability Letters, Elsevier, vol. 78(12), pages 1612-1618, September.
    12. Fabrice Gilles & Sabina Issehnane & Florent Sari, 2022. "Using short-term jobs as a way to find a regular job. What kind of role for local context?," TEPP Working Paper 2022-07, TEPP.
    13. repec:hal:spmain:info:hdl:2441/dambferfb7dfprc9m052g20qh is not listed on IDEAS
    14. Paulo M. D. C. Parente & Richard J. Smith, 2021. "Quasi‐maximum likelihood and the kernel block bootstrap for nonlinear dynamic models," Journal of Time Series Analysis, Wiley Blackwell, vol. 42(4), pages 377-405, July.
    15. Cornelia Lawson, 2013. "Academic Inventions Outside the University: Investigating Patent Ownership in the UK," Industry and Innovation, Taylor & Francis Journals, vol. 20(5), pages 385-398, July.
    16. Vipin Arora & Shuping Shi, 2016. "Nonlinearities and tests of asset price bubbles," Empirical Economics, Springer, vol. 50(4), pages 1421-1433, June.
    17. Luiz Paulo Fávero & Joseph F. Hair & Rafael de Freitas Souza & Matheus Albergaria & Talles V. Brugni, 2021. "Zero-Inflated Generalized Linear Mixed Models: A Better Way to Understand Data Relationships," Mathematics, MDPI, vol. 9(10), pages 1-28, May.
    18. Da Fonseca José & Grasselli Martino & Ielpo Florian, 2014. "Estimating the Wishart Affine Stochastic Correlation Model using the empirical characteristic function," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 18(3), pages 1-37, May.
    19. Hansen, Lars Peter & Heaton, John & Luttmer, Erzo G J, 1995. "Econometric Evaluation of Asset Pricing Models," The Review of Financial Studies, Society for Financial Studies, vol. 8(2), pages 237-274.
    20. Das, Marcel & van Soest, Arthur, 1999. "A panel data model for subjective information on household income growth," Journal of Economic Behavior & Organization, Elsevier, vol. 40(4), pages 409-426, December.
    21. Gillespie, Colin S., 2015. "Fitting Heavy Tailed Distributions: The poweRlaw Package," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 64(i02).

    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:174:y:2022:i:c:s0167947322000172. 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.