A Normality Test for Multivariate Dependent Samples

Most normality tests in the literature are performed for scalar and independent samples. Thus, they become unreliable when applied to colored processes, hampering their use in realistic scenarios. We focus on Mardia’s multivariate kurtosis, derive closed-form expressions of its asymptotic distribution for statistically dependent samples, under the null hypothesis of normality. Included experiments illustrate, by means of copulas, that it does not suffice to test a one-dimensional marginal to conclude normality. The proposed test also exhibits good properties on other typical scenarios, such as the detection of a non-Gaussian process in the presence of an additive Gaussian noise.

[1]  Alicia Nieto-Reyes,et al.  A random-projection based test of Gaussianity for stationary processes , 2014, Comput. Stat. Data Anal..

[2]  Andrzej Cichocki,et al.  Adaptive blind signal and image processing , 2002 .

[3]  Norbert Henze,et al.  Invariant tests for multivariate normality: a critical review , 2002 .

[4]  H. Cramér A contribution to the theory of statistical estimation , 1946 .

[5]  M. Hinich Testing for Gaussianity and Linearity of a Stationary Time Series. , 1982 .

[6]  P. McCullagh Tensor Methods in Statistics , 1987 .

[7]  E. S. Pearson,et al.  Tests for departure from normality: Comparison of powers , 1977 .

[8]  David S. Moore,et al.  The Effect of Dependence on Chi Squared Tests of Fit , 1982 .

[9]  A. Afifi,et al.  On Tests for Multivariate Normality , 1973 .

[10]  K. Mardia Measures of multivariate skewness and kurtosis with applications , 1970 .

[11]  Michèle Basseville,et al.  Detection of abrupt changes: theory and application , 1993 .

[12]  Wenhao Yu,et al.  Supplementary material , 2015 .

[13]  J. L. Warner,et al.  Methods for Assessing Multivariate Normality , 1973 .

[14]  Marius Hofert,et al.  Sampling Archimedean copulas , 2008, Comput. Stat. Data Anal..

[15]  S. Shapiro,et al.  A Comparative Study of Various Tests for Normality , 1968 .

[16]  T. W. Epps Testing That a Stationary Time Series is Gaussian , 1987 .

[17]  Eric Moulines,et al.  Testing that a multivariate stationary time-series is Gaussian , 1992, [1992] IEEE Sixth SP Workshop on Statistical Signal and Array Processing.

[18]  D. S. Moore,et al.  A CHI-SQUARE STATISTIC WITH RANDOM CELL BOUNDARIES' , 1971 .

[19]  Ralph B. D'Agostino,et al.  Tests for Departure from Normality , 1973 .

[20]  Yossef Steinberg,et al.  On tests for normality , 1992, IEEE Trans. Inf. Theory.

[21]  T. Gasser Goodness-of-fit tests for correlated data , 1975 .

[22]  L. Shenton,et al.  Omnibus test contours for departures from normality based on √b1 and b2 , 1975 .

[23]  D. Pfeffermann,et al.  Small area estimation , 2011 .

[24]  Pierre Comon,et al.  Handbook of Blind Source Separation: Independent Component Analysis and Applications , 2010 .