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The complex multinormal distribution, quadratic forms in complex random vectors and an omnibus goodness-of-fit test for the complex normal distribution

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  • Gilles Ducharme
  • Pierre Lafaye de Micheaux
  • Bastien Marchina

Abstract

This paper first reviews some basic properties of the (noncircular) complex multinormal distribution and presents a few characterizations of it. The distribution of linear combinations of complex normally distributed random vectors is then obtained, as well as the behavior of quadratic forms in complex multinormal random vectors. We look into the problem of estimating the complex parameters of the complex normal distribution and give their asymptotic distribution. We then propose a virtually omnibus goodness-of-fit test for the complex normal distribution with unknown parameters, based on the empirical characteristic function. Monte Carlo simulation results show that our test behaves well against various alternative distributions. The test is then applied to an fMRI data set and we show how it can be used to “validate” the usual hypothesis of normality of the outside-brain signal. An R package that contains the functions to perform the test is available from the authors. Copyright The Institute of Statistical Mathematics, Tokyo 2016

Suggested Citation

  • Gilles Ducharme & Pierre Lafaye de Micheaux & Bastien Marchina, 2016. "The complex multinormal distribution, quadratic forms in complex random vectors and an omnibus goodness-of-fit test for the complex normal distribution," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 68(1), pages 77-104, February.
    • Handle: RePEc:spr:aistmt:v:68:y:2016:i:1:p:77-104
    DOI: 10.1007/s10463-014-0486-5
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    References listed on IDEAS

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    1. Norbert Henze, 2002. "Invariant tests for multivariate normality: a critical review," Statistical Papers, Springer, vol. 43(4), pages 467-506, October.
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    Cited by:

    1. Amengual, Dante & Carrasco, Marine & Sentana, Enrique, 2020. "Testing distributional assumptions using a continuum of moments," Journal of Econometrics, Elsevier, vol. 218(2), pages 655-689.
    2. 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.
    3. Fan, Yanan & de Micheaux, Pierre Lafaye & Penev, Spiridon & Salopek, Donna, 2017. "Multivariate nonparametric test of independence," Journal of Multivariate Analysis, Elsevier, vol. 153(C), pages 189-210.

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