1 Introduction
The popularity of normality, being an underlying assumption to many forecasting and inference models, has led to the development of many procedures aiming at testing the hypothesis of Gaussianity; especially in the case of independent (univariate or multivariate) samples, see the surveys of [1] and [2].
Despite the practical importance of having statistically dependent variables, the majority of the tests are derived under the assumption that the latter are identically distributed and independent, see [3], [4] and [5] to cite few of an extensive literature. There has been considerable efforts to test the goodnessoffit of stationary colored processes, such as the Epps test [6]
based on the characteristic function, the Lobato Velasco’s (LV)
[7]modification of the test statistic proposed by
[4], and a test statistic that uses 1D random projections [8] to upgrade Epps and LV procedures.The lack of testing procedures for dependent samples is exacerbated in the multivariate setting. Available tests are scarce, and a powerful test like the bispectrum proposed in [9] suffers from severe drawbacks in practice. For this reason, we have recently proposed a computationally efficient test for multivariate timeseries [10], specifically in the bivariate case. Our work is at the crossroad between all these works: that is, those on multivariate procedures, i.e testing the joint normality and those derived for colored processes. The questions addressed in this communication are:

What is the impact of taking into account the statistical dependence among time samples?
Our main contributions may be summarized as follows:

The use of a joint normality test applied to twodimensional projections of variate colored processes.

Copulabased computer experiments confirm that testing twodimensional random projections is far better than their scalar counterparts applied on onedimensional projections. This observation is more noticeable for colored processes.
Organisation of the paper. We first formulate the normality test as a binary hypothesis test in Section 2. The test statistic [10] is defined in Sections 3 and 4
; its asymptotic mean and variance for different scenarios (multivariate) i.i.d and scalar or bivariate colored processes are stated. Sections
5 and 6 are dedicated to computer experiments.2 Problem formulation
Let be a
variate stochastic process. In this paper, the processes are considered zeromean stationary with finite moments up to order
. Let be the covariance function whose entries are . Also denote . It is also assumed that is strongmixing so that the series converges absolutely. The problem is formulated as:Problem P1: Given a sample of size of , , test
(1) where variables are identically distributed, but not statistically independent.
This normality test belongs to the class of tests without alternative. In this framework, a single parameter defines the nominal level of the test:
(2) 
For
, a standard measure of the gap from normality is the estimated Kurtosis:
(3) 
Following Mardia’s definition, we assume the extension of this measure to multivariate processes.
2.1 Mardia’s Kurtosis
The multivariate counterpart of the empirical kurtosis takes the form of:
(4) 
with the covariance matrix. Usually, this quantity is unknown and should be estimated on observations.
Our final test statistic takes the form:
(5) 
with
Theorem 2.1
[5] Let be i.i.d. of dimension
. Then under the null hypothesis
, is asymptotically normal, with mean and variance .Thus, we can test normality by measuring the normalized gap under :
(6) 
with means distributed as, and
denotes the univariate standard normal distribution. We reject the null hypothesis
at a significance level if:where
denotes the cumulative distribution function (cdf) of the standard normal distribution. A similar theorem without assuming independence among samples has been devised in
[10] for bivariate statistically dependent processes. For the rest of the paper, Mardia’s test statistic will be denoted to distinguish it from the tests assuming statistical dependence.3 Mean and variance of for a scalar colored process
In the case of scalar colored samples, the expressions of mean and variance of kurtosis are [10]:
(7) 
(8) 
The dependence between time samples is taken into account in the terms . Interestingly, if , the equations above reduce to the i.i.d case.
4 Mean and variance of in the bivariate case
In the bivariate case, expressions become rapidly much more complicated but we can still write them explicitly [10], as reported below:
(9) 
(10) 
In the above equations, two kinds of dependence appear: socalled spatial crossvariate dependence , and the dependence between timesamples, . Due to their length, expressions of and are not explicited here and can be found in [10].
5 Computer Experiments
Illustration on copula.
Our goal is to generate colored multivariate nonGaussian timeseries, whose marginals are Gaussian to make the problem more difficult. With this goal, we chose to use Archimedean copula for their ease to generate in dimension .
Definition 5.1
A dimensional copula is called Archimedean if it allows the representation:
(11) 
for some Archimedean generator and its inverse : ).
The parameter of the copula is related to Kendall rank correlation coefficient. Thus, it controls the spatial dependence between variables. In order to introduce time dependency between samples, an AR filter is applied on each marginal before constructing copula (this preserves normality). This leads to the following algorithm:
Sampling an Archimedean copula.

Sample i.i.d ,

Correlate ’s using a first order autoregressive filter:
Note that the first samples are dropped to alleviate startup effects ().

Sample where denotes the inverse LaplaceStieltjes transform of

Return (), where

Transform to obtain Gaussian standard marginals as the following:
(12)
The above algorithm is a slight modification to the one due to Mashall, Olkin (1988)[12]. In the remainder of this paper, we precisely use Gumbel () and Clayton () copula.
Lowdimensional projection.
We study the performance of the proposed test statistic on a lowdimensional (either or ) projection of the initial variate data. For a given copula , we carried out the following simulations:

Given one set of bivariate observations () of total length , they are projected
times onto the arbitrary vector
with coordinates ().is sampled from a uniform distribution on
denoted . Fig.2 shows an illustrative example with two copulas. 
Given one set of trivariate observations of total length , the points are projected arbitrarily times onto the plane defined by two angles (the angle between the axis and the new plane) and (measured between the axis and the vector inside the plane). Fig. 2 gives two illustrative examples of this procedure.
For each projection, we measure the values returned by the test statistic. Each value is compared with the level of the test ; if inferior to it the test rejects the null hypothesis of normality. The empirical rejection rates, summarized in the following tables are computed as .
6 Main results
We report the empirical rejection rate of each test statistic mentioned in Tables 1, 2, 3 and 4 and comment on the results for each scenario mentioned in the top left cell of each table. The results are reported for two significance levels and .
Scalar projection

and perform very poorly when used on arbitrary onedimensional projections of the Gumbel copula. The test power does not surpass .

For the Clayton copula, whose tails are asymmetric, the test has a better power than the Gumbel copula. Although this observation is less demonstrative, we keep those results to further compare them with the bivariate test statistic.

Since we only use firstorder autoregressive filters, there is no substantial difference in the performance of compared to ; this comparison is not of interest to us, because the bias induced by using tests assuming independence on colored processes has already been observed and studied in the literature [13] and [14].
However, it is interesting to compare Tables 1 and 2; we see that the overall performance of the test statistics tends to decrease when marginals are timecorrelated. 
Table 3 shows performances obtained with when applied to colored processes. Contrary to , performances do not decrease with timecorrelation. Furthermore, the power of the 2D test based on is not affected by a rotation in the plane (implemented by two scalar projections onto two orthogonal axes). This is illustrated by Table 3, which reports the results averaged over 5000 random rotations.
Bivariate projection

One would expect the same problem of misdetections to occur when projecting trivariate observations sampled from Gumbel copula. Yet, in Table 4 we show that the joint normality, even on a low representation of the data, is able to detect the nonGaussianity of the process.



Gumbel  0.1250  0.1328  0.1242  0.1316  
Clayton  0.661  0.72  0.652  0.713 



Gumbel  0.2134  0.2510  0.2082  0.2406  
Clayton  0.717  0.76  0.701  0.752 



Gumbel  0.9516  0.9574  
Clayton  0.9701  0.9882 



Gumbel  0.9492  0.9556  
Clayton  0.8540  0.87 
7 Concluding remarks
This study demonstrates, on one hand, that testing the joint normality of a twodimensional projection yields a noticeable increase in the power of the test to detect departure from joint normality, even in the most pathological scenario of the Gumbel copula. On the other hand, when data are additionally timecorrelated, the overall power of scalar tests tends to decrease. By assuming both spatial and temporal dependence, our bivariate test stands out from other existing multivariate tests that assume independence.
Future studies will be carried out to validate the performance of this statistic on real higher dimensional data.
References
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[5]
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