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Integral transform methods in goodness-of-fit testing, II: the Wishart distributions

Author

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  • Elena Hadjicosta

    (Pennsylvania State University)

  • Donald Richards

    (Pennsylvania State University)

Abstract

We initiate the study of goodness-of-fit testing for data consisting of positive definite matrices. Motivated by the appearance of positive definite matrices in numerous applications, including factor analysis, diffusion tensor imaging, volatility models for financial time series, wireless communication systems, and polarimetric radar imaging, we apply the method of Hankel transforms of matrix argument to develop goodness-of-fit tests for Wishart distributions with given shape parameter and unknown scale matrix. We obtain the limiting null distribution of the test statistic and a corresponding covariance operator, show that the eigenvalues of the operator satisfy an interlacing property, and apply our test to some financial data. We establish the consistency of the test against a large class of alternative distributions and derive the asymptotic distribution of the test statistic under a sequence of contiguous alternatives. We obtain the Bahadur and Pitman efficiency properties of the test statistic and establish a modified version of Wieand’s condition.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:aistmt:v:72:y:2020:i:6:d:10.1007_s10463-019-00737-z
    DOI: 10.1007/s10463-019-00737-z
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    References listed on IDEAS

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    4. Manabu Asai & Michael McAleer & Jun Yu, 2006. "Multivariate Stochastic Volatility," Microeconomics Working Papers 22058, East Asian Bureau of Economic Research.
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    6. Siriteanu, Constantin & Kuriki, Satoshi & Richards, Donald & Takemura, Akimichi, 2016. "Chi-square mixture representations for the distribution of the scalar Schur complement in a noncentral Wishart matrix," Statistics & Probability Letters, Elsevier, vol. 115(C), pages 79-87.
    7. Severini,Thomas A., 2005. "Elements of Distribution Theory," Cambridge Books, Cambridge University Press, number 9780521844727, January.
    8. Ronald W. Butler, 1998. "Generalized Inverse Gaussian Distributions and their Wishart Connections," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 25(1), pages 69-75, March.
    9. Gourieroux, Christian & Sufana, Razvan, 2010. "Derivative Pricing With Wishart Multivariate Stochastic Volatility," Journal of Business & Economic Statistics, American Statistical Association, vol. 28(3), pages 438-451.
    10. Michael Browne, 1968. "A comparison of factor analytic techniques," Psychometrika, Springer;The Psychometric Society, vol. 33(3), pages 267-334, September.
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    Cited by:

    1. Armine Bagyan & Donald Richards, 2023. "Hoffmann-Jørgensen Inequalities for Random Walks on the Cone of Positive Definite Matrices," Journal of Theoretical Probability, Springer, vol. 36(2), pages 1181-1202, June.
    2. Žikica Lukić & Bojana Milošević, 2024. "Change-point analysis for matrix data: the empirical Hankel transform approach," Statistical Papers, Springer, vol. 65(9), pages 5955-5980, December.
    3. Žikica Lukić & Bojana Milošević, 2024. "A novel two-sample test within the space of symmetric positive definite matrix distributions and its application in finance," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 76(5), pages 797-820, October.
    4. Ouimet, Frédéric, 2022. "A symmetric matrix-variate normal local approximation for the Wishart distribution and some applications," Journal of Multivariate Analysis, Elsevier, vol. 189(C).

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