IDEAS home Printed from https://ideas.repec.org/a/spr/aistmt/v69y2017i5d10.1007_s10463-016-0567-8.html
   My bibliography  Save this article

The limit distribution of weighted $$L^2$$ L 2 -goodness-of-fit statistics under fixed alternatives, with applications

Author

Listed:
  • L. Baringhaus

    (Leibniz Universität Hannover)

  • B. Ebner

    (Institut für Stochastik)

  • N. Henze

    (Institut für Stochastik)

Abstract

We present a general result on the limit distribution of weighted one- and two-sample $$L^2$$ L 2 -goodness-of-fit test statistics of some hypothesis $$H_0$$ H 0 under fixed alternatives. Applications include an approximation of the power function of such tests, asymptotic confidence intervals of the distance of an underlying distribution with respect to the distributions under $$H_0$$ H 0 , and an asymptotic equivalence test that is able to validate certain neighborhoods of $$H_0$$ H 0 .

Suggested Citation

  • L. Baringhaus & B. Ebner & N. Henze, 2017. "The limit distribution of weighted $$L^2$$ L 2 -goodness-of-fit statistics under fixed alternatives, with applications," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(5), pages 969-995, October.
    • Handle: RePEc:spr:aistmt:v:69:y:2017:i:5:d:10.1007_s10463-016-0567-8
    DOI: 10.1007/s10463-016-0567-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10463-016-0567-8
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10463-016-0567-8?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. N. Henze, 1990. "An approximation to the limit distribution of the epps-pulley test statistic for normality," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 37(1), pages 7-18, December.
    2. Meintanis, Simos G. & Tsionas, Efthimios, 2010. "Testing for the generalized normal-Laplace distribution with applications," Computational Statistics & Data Analysis, Elsevier, vol. 54(12), pages 3174-3180, December.
    3. Henze, Norbert & Wagner, Thorsten, 1997. "A New Approach to the BHEP Tests for Multivariate Normality," Journal of Multivariate Analysis, Elsevier, vol. 62(1), pages 1-23, July.
    4. Henze, N. & Klar, B. & Meintanis, S. G., 2003. "Invariant tests for symmetry about an unspecified point based on the empirical characteristic function," Journal of Multivariate Analysis, Elsevier, vol. 87(2), pages 275-297, November.
    5. L. Baringhaus & D. Kolbe, 2015. "Two-sample tests based on empirical Hankel transforms," Statistical Papers, Springer, vol. 56(3), pages 597-617, August.
    6. Bücher, Axel & Dette, Holger, 2010. "Some comments on goodness-of-fit tests for the parametric form of the copula based on L2-distances," Journal of Multivariate Analysis, Elsevier, vol. 101(3), pages 749-763, March.
    7. Norbert Henze, 2002. "Invariant tests for multivariate normality: a critical review," Statistical Papers, Springer, vol. 43(4), pages 467-506, October.
    8. Nora Gürtler & Norbert Henze, 2000. "Goodness-of-Fit Tests for the Cauchy Distribution Based on the Empirical Characteristic Function," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(2), pages 267-286, June.
    9. Marie Hušková & Simos Meintanis, 2008. "Tests for the multivariate -sample problem based on the empirical characteristic function," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(3), pages 263-277.
    10. Henze, N. & Klar, B. & Zhu, L. X., 2005. "Checking the adequacy of the multivariate semiparametric location shift model," Journal of Multivariate Analysis, Elsevier, vol. 93(2), pages 238-256, April.
    11. Ludwig Baringhaus & Norbert Henze, 1991. "A class of consistent tests for exponentiality based on the empirical Laplace transform," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 43(3), pages 551-564, September.
    12. Simos Meintanis & George Iliopoulos, 2003. "Tests of fit for the Rayleigh distribution based on the empirical Laplace transform," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 55(1), pages 137-151, March.
    13. Norbert Henze & Simos G. Meintanis, 2005. "Recent and classical tests for exponentiality: a partial review with comparisons," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 61(1), pages 29-45, February.
    14. Holger Dette & Axel Munk, 2003. "Some Methodological Aspects of Validation of Models in Nonparametric Regression," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 57(2), pages 207-244, May.
    15. Leucht, Anne, 2012. "Characteristic function-based hypothesis tests under weak dependence," Journal of Multivariate Analysis, Elsevier, vol. 108(C), pages 67-89.
    16. Gupta, A. K. & Henze, N. & Klar, B., 2004. "Testing for affine equivalence of elliptically symmetric distributions," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 222-242, February.
    17. Norbert Henze & Bernhard Klar, 2002. "Goodness-of-Fit Tests for the Inverse Gaussian Distribution Based on the Empirical Laplace Transform," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 54(2), pages 425-444, June.
    18. Baringhaus, L. & Henze, N., 1992. "A goodness of fit test for the Poisson distribution based on the empirical generating function," Statistics & Probability Letters, Elsevier, vol. 13(4), pages 269-274, March.
    19. Henze, Norbert, 1997. "Extreme smoothing and testing for multivariate normality," Statistics & Probability Letters, Elsevier, vol. 35(3), pages 203-213, October.
    20. Kanta Naito, 1997. "On the asymptotic normality of the L 2 -Distance Class of Statistics with Estimated Parameters," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 8(3), pages 199-214, September.
    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. Steffen Betsch & Bruno Ebner, 2020. "Testing normality via a distributional fixed point property in the Stein characterization," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(1), pages 105-138, March.
    2. Norbert Henze & Celeste Mayer, 2020. "More good news on the HKM test for multivariate reflected symmetry about an unknown centre," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(3), pages 741-770, June.
    3. E. Bothma & J. S. Allison & I. J. H. Visagie, 2022. "New classes of tests for the Weibull distribution using Stein’s method in the presence of random right censoring," Computational Statistics, Springer, vol. 37(4), pages 1751-1770, September.
    4. M. Dolores Jiménez-Gamero, 2020. "Comments on: Tests for multivariate normality—a critical review with emphasis on weighted $$L^2$$ L 2 -statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(4), pages 893-897, December.
    5. Chen, Feifei & Jiménez–Gamero, M. Dolores & Meintanis, Simos & Zhu, Lixing, 2022. "A general Monte Carlo method for multivariate goodness–of–fit testing applied to elliptical families," Computational Statistics & Data Analysis, Elsevier, vol. 175(C).
    6. Norbert Henze & María Dolores Jiménez‐Gamero, 2021. "A test for Gaussianity in Hilbert spaces via the empirical characteristic functional," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(2), pages 406-428, June.
    7. Sangyeol Lee & Simos G. Meintanis & Minyoung Jo, 2019. "Inferential procedures based on the integrated empirical characteristic function," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 103(3), pages 357-386, September.
    8. Bruno Ebner & Norbert Henze, 2020. "Rejoinder on: Tests for multivariate normality—a critical review with emphasis on weighted $$L^2$$ L 2 -statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(4), pages 911-913, December.
    9. L. Baringhaus & D. Gaigall & J. P. Thiele, 2018. "Statistical inference for $$L^2$$ L 2 -distances to uniformity," Computational Statistics, Springer, vol. 33(4), pages 1863-1896, December.
    10. Steffen Betsch & Bruno Ebner, 2019. "A new characterization of the Gamma distribution and associated goodness-of-fit tests," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(7), pages 779-806, October.
    11. Norbert Henze & Pierre Lafaye De Micheaux & Simos G. Meintanis, 2022. "Tests for circular symmetry of complex-valued random vectors," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(2), pages 488-518, June.
    12. Bruno Ebner & Norbert Henze, 2020. "Tests for multivariate normality—a critical review with emphasis on weighted $$L^2$$ L 2 -statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(4), pages 845-892, December.
    13. Norbert Henze & María Dolores Jiménez-Gamero, 2019. "A new class of tests for multinormality with i.i.d. and garch data based on the empirical moment generating function," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 499-521, June.
    14. Philip Dörr & Bruno Ebner & Norbert Henze, 2021. "Testing multivariate normality by zeros of the harmonic oscillator in characteristic function spaces," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(2), pages 456-501, June.
    15. Elena Hadjicosta & Donald Richards, 2020. "Integral transform methods in goodness-of-fit testing, I: the gamma distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(7), pages 733-777, October.
    16. Elena Hadjicosta & Donald Richards, 2020. "Integral transform methods in goodness-of-fit testing, II: the Wishart distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(6), pages 1317-1370, December.
    17. Philip Dörr & Bruno Ebner & Norbert Henze, 2021. "A new test of multivariate normality by a double estimation in a characterizing PDE," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(3), pages 401-427, April.

    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. Bojana Milošević & Marko Obradović, 2016. "New class of exponentiality tests based on U-empirical Laplace transform," Statistical Papers, Springer, vol. 57(4), pages 977-990, December.
    2. Meintanis, Simos & Swanepoel, Jan, 2007. "Bootstrap goodness-of-fit tests with estimated parameters based on empirical transforms," Statistics & Probability Letters, Elsevier, vol. 77(10), pages 1004-1013, June.
    3. Chen, Feifei & Jiménez–Gamero, M. Dolores & Meintanis, Simos & Zhu, Lixing, 2022. "A general Monte Carlo method for multivariate goodness–of–fit testing applied to elliptical families," Computational Statistics & Data Analysis, Elsevier, vol. 175(C).
    4. Philip Dörr & Bruno Ebner & Norbert Henze, 2021. "Testing multivariate normality by zeros of the harmonic oscillator in characteristic function spaces," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(2), pages 456-501, June.
    5. Philip Dörr & Bruno Ebner & Norbert Henze, 2021. "A new test of multivariate normality by a double estimation in a characterizing PDE," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(3), pages 401-427, April.
    6. Norbert Henze & Celeste Mayer, 2020. "More good news on the HKM test for multivariate reflected symmetry about an unknown centre," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(3), pages 741-770, June.
    7. Wanfang Chen & Marc G. Genton, 2023. "Are You All Normal? It Depends!," International Statistical Review, International Statistical Institute, vol. 91(1), pages 114-139, April.
    8. Bruno Ebner & Norbert Henze, 2020. "Tests for multivariate normality—a critical review with emphasis on weighted $$L^2$$ L 2 -statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(4), pages 845-892, December.
    9. Chen, Feifei & Meintanis, Simos G. & Zhu, Lixing, 2019. "On some characterizations and multidimensional criteria for testing homogeneity, symmetry and independence," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 125-144.
    10. M. D. Jiménez-Gamero & J. L. Moreno-Rebollo & J. A. Mayor-Gallego, 2018. "On the estimation of the characteristic function in finite populations with applications," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(1), pages 95-121, March.
    11. M. Jiménez Gamero, 2014. "On the empirical characteristic function process of the residuals in GARCH models and applications," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 409-432, June.
    12. Norbert Henze & María Dolores Jiménez-Gamero, 2019. "A new class of tests for multinormality with i.i.d. and garch data based on the empirical moment generating function," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 499-521, June.
    13. Tenreiro, Carlos, 2011. "An affine invariant multiple test procedure for assessing multivariate normality," Computational Statistics & Data Analysis, Elsevier, vol. 55(5), pages 1980-1992, May.
    14. Steffen Betsch & Bruno Ebner, 2020. "Testing normality via a distributional fixed point property in the Stein characterization," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(1), pages 105-138, March.
    15. Norbert Henze & Jaco Visagie, 2020. "Testing for normality in any dimension based on a partial differential equation involving the moment generating function," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(5), pages 1109-1136, October.
    16. Baringhaus, Ludwig & Taherizadeh, Fatemeh, 2010. "Empirical Hankel transforms and its applications to goodness-of-fit tests," Journal of Multivariate Analysis, Elsevier, vol. 101(6), pages 1445-1457, July.
    17. Gerrit Lodewicus Grobler & Elzanie Bothma & James Samuel Allison, 2022. "Testing for the Rayleigh Distribution: A New Test with Comparisons to Tests for Exponentiality Based on Transformed Data," Mathematics, MDPI, vol. 10(8), pages 1-17, April.
    18. Meintanis, Simos G. & Ngatchou-Wandji, Joseph & Taufer, Emanuele, 2015. "Goodness-of-fit tests for multivariate stable distributions based on the empirical characteristic function," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 171-192.
    19. Norbert Henze, 2002. "Invariant tests for multivariate normality: a critical review," Statistical Papers, Springer, vol. 43(4), pages 467-506, October.
    20. Simos Meintanis & Bojana Milošević & Marko Obradović, 2023. "Bahadur efficiency for certain goodness-of-fit tests based on the empirical characteristic function," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 86(7), pages 723-751, October.

    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:spr:aistmt:v:69:y:2017:i:5:d:10.1007_s10463-016-0567-8. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.