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A test for Gaussianity in Hilbert spaces via the empirical characteristic functional

Author

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  • Norbert Henze
  • María Dolores Jiménez‐Gamero

Abstract

Let X1,X2,… be independent and identically distributed random elements taking values in a separable Hilbert space ℍ. With applications for functional data in mind, ℍ may be regarded as a space of square‐integrable functions, defined on a compact interval. We propose and study a novel test of the hypothesis H0 that X1 has some unspecified nondegenerate Gaussian distribution. The test statistic Tn = Tn(X1,…,Xn) is based on a measure of deviation between the empirical characteristic functional of X1,…,Xn and the characteristic functional of a suitable Gaussian random element of ℍ. We derive the asymptotic distribution of Tn as n→∞ under H0 and provide a consistent bootstrap approximation thereof. Moreover, we obtain an almost sure limit of Tn and the limit distributions of Tn under fixed and contiguous alternatives to Gaussianity. Simulations show that the new test is competitive with respect to the hitherto few competitors available.

Suggested Citation

  • 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.
    • Handle: RePEc:bla:scjsta:v:48:y:2021:i:2:p:406-428
    DOI: 10.1111/sjos.12470
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    References listed on IDEAS

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    1. Natalie Neumeyer & Miguel A. Delgado & Lajos Horváth & Simos Meintanis & Emanuele Taufer & Lixing Zhu, 2021. "4th Workshop on Goodness‐of‐Fit, Change‐Point, and Related Problems, Trento, 2019," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(2), pages 371-374, June.
    2. Meintanis, Simos G. & Hušková, Marie & Hlávka, Zdeněk, 2022. "Fourier-type tests of mutual independence between functional time series," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    3. Petr Čoupek & Viktor Dolník & Zdeněk Hlávka & Daniel Hlubinka, 2024. "Fourier approach to goodness-of-fit tests for Gaussian random processes," Statistical Papers, Springer, vol. 65(5), pages 2937-2972, July.

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