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FREQUENCY DOMAIN TESTS OF MULTIVARIATE GAUSSIANITY AND LINEARITY
Author(s) -
Wong Woon
Publication year - 1997
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/1467-9892.00045
Subject(s) - mathematics , independent and identically distributed random variables , multivariate statistics , linearity , multivariate random variable , gaussian process , series (stratigraphy) , multivariate normal distribution , gaussian , statistics , random variable , paleontology , physics , quantum mechanics , biology
A stationary multivariate time series { X t } is defined as linear if it can be written in the form X t = ∑ ∞ j =−∞ A j e t − j where A j are square matrices and e t are independent and identically distributed random vectors. If the e t } are normally distributed, then { X t is a multivariate Gaussian linear process. This paper is concerned with the testing of departures of a vector stationary process from multivariate Gaussianity and linearity using the bispectral approach. First the definition and properties of cumulants of random matrices are used to obtain the expressions for the higher‐order cumulant and spectral vectors of a linear vector process as defined above. Then it is shown that linearity of a vector process implies constancy of the modulus square of its normalized higher‐order spectra whereas the component of such a vector process does not necessarily have a linear representation. Finally, statistics for the testing of multivariate Gaussianity and linearity are proposed.