Let Z 1 , ⋯ , Z N be an independent sample from N p ( μ , Σ ) , p ≥ 2 , an multivariate normal distribution with mean vector μ , covariance matrix Σ , where

Z i is expressed by ( X i Y i ) , i = 1 , ⋯ , N , and Σ is a blocked matrix given by

Σ = ( Σ x Γ Γ ′ Σ y ) , (2)

with Γ being the cross-covariance matrix. In this section, we consider the problem of testing

H 0 : Γ = Γ 0     versus     H 1 : Γ ≠ Γ 0 .

The test statistic is constructed based on the Frobenius norm of the difference between the sample cross-covariance matrix and the given matrix. In the derivation, no assumption on p, like n ≥ p or n < p , is required. Since the Wishart distribution is not achieved here, we implement the derivation by the aid of parametric bootstrap scheme. We define the test statistic

T = N ‖ 1 N S x y − Γ 0 ‖ F 2 , (3)

where

S x y = ∑ i = 1 N ( X i − X ¯ ) ⋅ ( Y i − Y ¯ ) ′ , (4)

with X ¯ = N − 1 ∑ i = 1 N   X i and Y ¯ = N − 1 ∑ i = 1 N   Y i . In (3),

‖ 1 N S x y − Γ 0 ‖ F 2 = tr [ ( 1 N S x y − Γ 0 ) T ( 1 N S x y − Γ 0 ) ] , which can detect any deviation

of cross-covariance from the pre-specified matrix Γ 0 .

Let Z 1 , Z 2 , ⋯ , Z N be an independent sample that are drawn from N p ( μ , Σ ) . Σ ^ x and Σ ^ y are the estimator of the parameters Σ x and Σ y . Suppose that the pseudo data set Z 1 ∗ , ⋯ , Z N ∗ was resample from N p ( μ , Σ ∗ ) , where

Σ ∗ = ( Σ ^ x Γ Γ ′ Σ ^ y ) , (5)

and Z i ∗ is expressed by ( X i ∗ Y i ∗ ) , i = 1 , ⋯ , N . The bootstrap statistic is defined as

T ∗ = N ‖ 1 N S x y ∗ − Γ 0 ‖ F 2 , (6)

where

S x y ∗ = ∑ i = 1 N ( X i ∗ − X ¯ ∗ ) ⋅ ( Y i ∗ − Y ¯ ∗ ) ′ , (7)

with X ¯ ∗ = N − 1 ∑ i = 1 N   X i ∗ and Y ¯ ∗ = N − 1 ∑ i = 1 N   Y i ∗ .

To study the bootstrap approximation of the null distribution, we need the following conditions:

Condition (A1) Let F ˜ be the cumulative distribution function (cdf) of the bootstrap statistic T ∗ , F M be the empirical distribution function of the bootstrap sample T 1 ∗ , ⋯ , T M ∗ . Let

W F ˜ M ( t ) = M { F M ( t ) − F ˜ ( t ) } . (8)

We assume the value of the equality (8) vanishes as t goes to infinity.

Condition (A2) As M tends to infinity, W F ˜ M converges weakly to B ( F ) in distribution, provided that sup − ∞ < x < + ∞ | F ˜ ( x ) − F ( x ) | → 0 a.s., where F denotes the cdf of the statistic T and B is the Brownian bridge on [0,1].

Theorem 1. Suppose F is nondegenerate. Suppose also that 0 < α < 1 be the nominal size of the test. Under conditions (A1) and (A2), if c ( F ˜ ) satisfies

P { M 1 / 2 sup x | F M ( x ) − F ˜ ( x ) | ≤ c ( F ˜ ) } → 1 − α , (9)

then

P { M 1 / 2 sup x | F ˜ ( x ) − F ( x ) | ≤ c ( F ˜ ) } → 1 − α . (10)

The result (9) is almost immediate from Corollary 4.2 of Bickel et al. [17] . By Lemma 8.11 of Bickel et al. [17] , we obtain that sup x | B ( F ( x ) ) | has a continuous distribution, c ( F ˜ ) converges to the ( 1 − α ) -quantile of the law of sup x | B ( F ( x ) ) | .

In order to carry out the parametric bootstrap procedure for the proposed test, we present the following simulation steps. The bootstrap p-value is approximated by the following procedure.

Step 1. Calculate an observation T o b s of statistic T.

Step 2. Estimate the covariance matrix Σ x and Σ y by sample covariance matrices, say Σ ^ x and Σ ^ y .

Step 3. Resample from N p ( μ , Σ ∗ ) and calculate the value of bootstrap statistic T ∗ , where Σ ∗ is defined in (5).

Step 4. Repeat Step 3 M times, and compute the p-value by p-value =

1 M ∑ i = 1 M I { T i ∗ > T o b s } , where I A denotes an indicator on the set A, which equals 1

when A occurs, and 0 otherwise.

We briefly describe the tests which are compared in the current paper, the modified LR test [18] , and the test of Nagao [8] . These two tests are used to test

H 0 : Σ = Σ 0     versus     H 1 : Σ ≠ Σ 0 .

Let Z 1 , ⋯ , Z N be an independent sample from N p ( μ , Σ ) , the modified LR test statistic is based on

λ 1 ∗ = e 1 2 p n ( det ( S Σ 0 ) e − tr ( S Σ 0 − 1 ) ) 1 2 n , (11)

where S = ( 1 / n ) A , n = N − 1 , A = ∑ i N ( Z i − Z ¯ ) ( Z i − Z ¯ ) ′ , and Z ¯ = N − 1 ∑ i = 1 N   Z i . As is known, A ~ W p ( N − 1, Σ ) , where the W p ( N − 1, Σ ) stands for p-dimension Wishart distribution with N − 1 degrees of freedom and covariance matrix Σ . Anderson [19] derived the limiting distribution of the modified LR test statistic − 2 log λ 1 ∗ with the help of Wishart distribution, when Σ = I p , − 2 log λ 1 ∗ is

asymptotically distributed as χ 2 ( 1 2 p ( p + 1 ) ) , where log a denotes the

logarithmic function based on natural logarithm e. Then, Nagao [8] proposed a test statistic T 1 (defined in (1)) which can be regarded as a measure of departure from the null hypothesis.

In what follows, we propose the statistic T (3) to test

H 0 : Γ = O p 2     versus     H 1 : Γ ≠ O p 2 . (12)

So far, no test methods are available for this problem. Thus, we choose statistics − 2 log λ 1 ∗ and T 1 to test

H 0 : Σ = I p     versus     H 1 : Σ ≠ I p , (13)

for the comparative study, where the cross-covariance matrix in Σ is equal to O p 2 . Testing the structure of the covariance matrix can also detect the deviation of cross-covariance from the pre-specified matrix. For a pre-specified level of significance α ( 0 < α < 1 ) , the null hypothesis in (13) is rejected if

− 2 log λ 1 ∗ > χ 1 − α 2 ( 1 2 p ( p + 1 ) ) , (14)

or

T 1 > T 1 1 − α ( n ) , (15)

where T 1 1 − α ( n ) denotes the 1 − α quantile of the empirical distribution of statistic T 1 .

We employ simulation data to evaluate the performance of the proposed statistic T, statistics − 2 log λ 1 ∗ and T 1 when applied to test the hypotheses (12) and (13) at a significant level of α = 0.05 . Empirical sizes and powers of the proposed test are computed based on M = 250 resample times and 500 repetition times, and that of tests (14) and (15) are based on 500 Monte Carlo replications. In the simulation study we choose mean vector μ = O p , dimension p = 4 , and take sample size n ∈ { 50,100,200,400,800 } . The results are shown in Table 1.

Results in Table 1 show that each empirical type I error rate of above three different statistics is very close to the pre-specified nominal value. Also, the proposed test has higher empirical power than its counterpart tests (14) and (15). With increasing sample size, the change in empirical power of test (15) is barely noticeable while the performance of the proposed test improves significantly, which means statistic T 1 is not sensitive to the change of the cross elements of the covariance matrix while statistic T can detect any deviation of cross-covariance from the pre-specified matrix Γ 0 . Thus, when the variance in the covariance matrix is not concerned, we recommend that statistic T can be applied to testing the equality hypothesis about a cross-covariance matrix.

In this section, we employ simulation data to investigate whether the performance of proposed test is sensitive to the block matrices in the diagonal. For this purpose, we consider two choices of Σ : one is

Σ 1 = ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) ,

and the other is

Σ 2 = ( 1 0.5 0 0 0.5 1 0 0 0 0 1 0.5 0 0 0.5 1 ) .

We propose statistic T to test

H 0 : Γ = O 2     versus     H 1 : Γ ≠ O 2 . (16)

For the two covariance matrices mentioned above, we run a simulation with M = 250 resample times and 500 repetition times to obtain the empirical sizes and powers of the proposed test at significant level α = 0.01 , 0.05 , 0.1 , where we take sample size n ∈ { 64,128,256,512,1024 } and choose mean vector μ = O p . The results are shown in Table 2 and Table 3.

For each sample size n = 128 , 256 , 512 , 1024 and each nominal size

α = 0.01 , 0.05 , 0.1 , Table 2 shows the empirical rejection probabilities of the proposed test. We present simulation results of Σ 1 and Σ 2 in the first panel

and the second panel, respectively. We see that each empirical type I error rate of two different cases is close to their nominal sizes. For the nominal size α = 0.05 , Table 3 shows how the empirical rejection probability of the proposed test changes with respect to five different sample sizes n = 64 , 128 , 256 , 512 , 1024 . Although the block matrices in the diagonal are different, their empirical powers improve significantly with increasing sample size n. Last but not least, when n = 512 , both the empirical powers of above two cases reach the maximum. By our simulation experiments in Table 2 and Table 3, we find that the proposed test is not sensitive to the block matrices in the diagonal. It is due to that the proposed statistic T depends only on the sample cross-covariance matrix. Thus we can conclude that our proposed test still achieve good performance though the change of the variance components in the covariance matrix take place.

A formal statistical definition of pairwise time-reversibility is defined as follows.

Definition 1. A time series { x t } t ∈ ℤ is pairwise time reversible if for all positive integers k, the random vectors ( x t , x t + k ) T and ( x t , x t − k ) T have the same joint probability distributions.

Under this definition, one can show that pairwise time reversibility implies stationarity. Likewise, nonstationarity implies time irreversibility [14] . Clearly, { x t } is time reversible when { x t } is i.i.d. Thus, for the study of testing time-reversibility, the pairwise case is generally considered. For instance, Ramsey et al. [12] proposed a pairwise TR test statistic consists of a sample estimate of the symmetric-bicovariance function given by the difference between two bicovariances of ( x t , x t − k ) . Laterly, Chen et al. [11] considered the pairwise time-reversibility and proposed a new test aiming at the symmetrical distribution of ( x t − x t − k ) rather than moments. In the following, Dette et al. [16] briefly analyzed the pairwise time-reversibility of four different time series models by the aid of quantile-based spectral analysis.

In this section, we primarily focus on testing for pairwise time-reversibility and the test method is based on copula spectral density kernel (CSDK) proposed by Dette et al. [16] . Let { x t } t ∈ ℤ be a strictly stationary univariate process, the CSDK is defined as

f q τ 1 , q τ 2 ( ω ) = 1 2 π ∑ k = − ∞ ∞ γ k U ( τ 1 , τ 2 ) e − i k ω ,   ( τ 1 , τ 2 ) ∈ ( 0,1 ) 2 , (17)

where q τ i is the τ i -quantile of the marginal distribution of the process { x t } t ∈ ℤ , i.e. q τ i = F − 1 ( τ i ) , i = 1 , 2 . The γ k U ( τ 1 , τ 2 ) (copula cross-covariance kernel) of lag k ∈ ℤ is also introduced by Dette et al. [16] which is defined as

γ k U ( τ 1 , τ 2 ) : = Cov ( I ( 0, τ 1 ] ( U t ) , I ( 0, τ 2 ] ( U t − k ) ) , (18)

where U t : = F ( x t ) , F denotes one-dimensional marginal cumulative distribution function of the process { x t } t ∈ ℤ . Compared with traditional covariances, the concept of copula cross-covariance kernel is proper for describing a serial copula.

The collection of CSDKs for different ( τ 1 , τ 2 ) provides a full characterization of the copulas associated with the pairs ( x t , x t − k ) , and accounts for many important dynamic features of { x t } , such as changes in the conditional shape (skewness, kurtosis), time-irreversibility, or dependence in the extremes that the traditional second-order spectra cannot capture [20] .

In the sequel, we will concentrate on testing for pairwise time-reversibility. Obviously, we have ( x t , x t + k ) = d ( x t , x t − k ) for all k ∈ ℤ if and only if ℑ f q τ 1 , q τ 2 ( ω ) = 0 for all ω and all ( τ 1 , τ 2 ) ∈ [ 0,1 ] 2 . For the purpose of pairwise time-reversibility test, we consider the problem of testing

H 0 : ℑ f q τ 1 , q τ 2 ( ω ) = 0     versus     H 1 : ℑ f q τ 1 , q τ 2 ( ω ) ≠ 0, (19)

for all ω ∈ ( 0, π ) . The method introduced by Dette et al. [16] for estimating the f q τ 1 , q τ 2 ( ω ) is first to calculate the rank-based Laplace periodogram (RLP) and then to smooth it to obtain the consistent estimator. Let { x 1 , ⋯ , x n } be the observation from a strictly stationary univariate process { x t } . Like Dette et al. [16] , we define b ^ by

( a ^ n , R τ ( ω j , n ) , b ^ n , R τ ( ω j , n ) ) : = Argmin a , b T ∈ ℝ 3 ∑ t = 1 n     ρ τ ( n − 1 R t ( n ) − ( a , b T ) c t ( ω j , n ) ) , (20)

where

ρ τ ( x ) : = x ( τ − I ( − ∞ ,0 ] ( x ) ) , (21)

is the so-called check function [21] , c t ( ω j , n ) : = ( 1, cos ( t ω j , n ) , sin ( t ω j , n ) ) T , ω j , n ∈ F n : = { 2 π j / n | j = 1, ⋯ , ⌊ n − 1 2 ⌋ − 1, ⌊ n − 1 2 ⌋ } . We extend the definition of b ^ n , R τ ( ω j , n ) to a piecewise constant function on Ω n : = ( ω j , n − 2π n , ω j , n + 2π n ] as follows:

b ^ n , R τ ( ω ) = b ^ n , R τ ( ω j , n ) ,     if   ω j , n − 2π n < ω < ω j , n + 2π n . (22)

Let τ = { τ 1 , τ 2 } T . We denote b ^ n , R τ ( ω ) = ( b ^ n , R τ 1 ( ω ) T , b ^ n , R τ 2 ( ω ) T ) T , where n b ^ n , R τ 1 ( ω ) = ( Z 11 ( ω ) , Z 12 ( ω ) ) T and n b ^ n , R τ 2 ( ω ) = ( Z 21 ( ω ) , Z 22 ( ω ) ) T . Then, under standard mixing conditions (Theorem 3.3 of Dette et al. [16] ), for ω ∈ ( 0, π ) , ( Z 11 ( ω ) , Z 12 ( ω ) , Z 21 ( ω ) , Z 22 ( ω ) ) T converges to a zero-mean real Gaussian distribution with covariance matrix

Σ 4 ( ω ) : = 4 π ( f q τ 1 , q τ 1 ( ω ) 0 ℜ f q τ 1 , q τ 2 ( ω ) ℑ f q τ 1 , q τ 2 ( ω ) 0 f q τ 1 , q τ 1 ( ω ) − ℑ f q τ 1 , q τ 2 ( ω ) ℜ f q τ 1 , q τ 2 ( ω ) ℜ f q τ 1 , q τ 2 ( ω ) − ℑ f q τ 1 , q τ 2 ( ω ) f q τ 2 , q τ 2 ( ω ) 0 ℑ f q τ 1 , q τ 2 ( ω ) ℜ f q τ 2 , q τ 2 ( ω ) 0 f q τ 2 , q τ 2 ( ω ) ) . (23)

Not all the entries in Σ 4 ( ω ) need to be concerned when testing hypotheses (19), so we consider the block covariance matrix Cov ( Z 11 ( ω ) , Z 22 ( ω ) ) ,

Σ 2 ( ω ) : = 4 π ( f q τ 1 , q τ 1 ( ω ) ℑ f q τ 1 , q τ 2 ( ω ) ℑ f q τ 1 , q τ 2 ( ω ) f q τ 2 , q τ 2 ( ω ) ) . (24)

Thus, for the test of hypotheses (19), we can transform the problem into one of testing the cross-covariance matrix in (24) being a one dimensional zero matrix. Here, we define random vectors X and Y as

X = sgn ( f ^ ˜ n , τ 1 , τ 1 ( ω ) ) Z 11 ( ω ) 4 π | f ^ ˜ n , τ 1 , τ 1 ( ω ) | , Y = sgn ( f ^ ˜ n , τ 2 , τ 2 ( ω ) ) Z 22 ( ω ) 4 π | f ^ ˜ n , τ 2 , τ 2 ( ω ) | ,

where the smoothed rank-based Laplace periodogram f ^ n , τ 1 , τ 2 ( ω ) denotes a consistent estimator of CSDK, and

sgn ( f ^ ˜ n , τ i , τ i ( ω ) ) = ( 1, if   f ^ ˜ n , τ i , τ i ( w ) > 0 , − 1, if   f ^ ˜ n , τ i , τ i ( w ) < 0 ,   i = 1,2 (25)

Let Z 1 , ⋯ , Z N be an independent sample, where Z i is expressed by ( X i Y i ) , i = 1 , ⋯ , ω . The test p-value is approximated by the previous bootstrap scheme.

Due to lack of the accurate values of CSDKs, it is difficult for us to evaluate our methodology by using the general models by simulation [20] . We consider two AR(1) models (models 1 and 2) with the form

x t = − 0.3 x t − 1 + ε t , (26)

since their CSDKs can be computed numerically. In model 1, ε t , t = 1 , 2 , ⋯ are independent N ( 0,1 ) -distributed random variables, while independent Student t-distributed with 1 degree of freedom in model 2. For model 1, the imaginary component of CSDK is vanishing, which reflects that the process is time-reversible; for model 2, there exists a time-irreversible impact of extreme values on the central ones [16] .

Recently, Dette et al. [16] and Kley et al. [22] proposed to estimate the CSDKs by smoothing RLP, which can be defined by quantile regression (QR) or clipped time series (CT). The finite sample performance of the smoothed RLP can be conducted using the R package quantspec [22] . This makes the method of smoothed (QR- or CT-based) RLP serve a good reference to calculate the test p-value. We take the smoothed QR-based RLP to cope with the pairwise TR test. For each generated pseudo-random time series, we computed the smoothed QR-based RLP using the Epanechnikov kernel and bandwidth b w = 0.07 . For each generated dataset and each pair of ( τ 1 , τ 2 ) , the test p-value is computed by the previous bootstrap procedure.

For each of those two models, we generated 512 and 1024 dataset with each containing pseudo-random time series of lengths n = 1024 and n = 2048 respectively. We set τ = ( 0.1 , 0.5 , 0.9 ) T . A boxplot that is drawn is based on 50 realizations of the log scale of the realized p-value. For each pairs ( τ 1 , τ 2 ) = ( 0.1 , 0.9 ) , ( 0.1 , 0.5 ) , ( 0.5 , 0.9 ) , the boxplots of two models are presented on the left, middle and right, respectively. For each boxplot, the median, extreme points and box shaped by lower and upper quartiles are marked in Figure 1 and Figure 2.

Next we discuss the simulation results in the case of an AR(1) process. We find from Figure 1 that the lower quartile of boxplots of two models is greater than log ( α ) at quantile pair ( 0.1,0.9 ) , which means the null hypothesis in (19) cannot be rejected, i.e. P ( x t ≤ q 0.1 , x t + k ≤ q 0.9 ) is approximately equal to

P ( x t ≤ q 0.9 , x t + k ≤ q 0.1 ) ,means that AR process with Gaussian innovations or t 1 distributed innovations is time-reversible. Also, when ( τ 1 , τ 2 ) = ( 0.1,0.5 ) or ( 0.5,0.9 ) , these observations also reflect the fact that AR process with Gaussian innovations are time-reversible. However, for t 1 distributed innovations, this phenomenon only takes place for the extreme quantiles ( τ 1 = 0.1 , τ 2 = 0.9 ), does not hold for τ 1 = 0.5 and τ 2 = 0.1 or τ 2 = 0.9 , one of the important reason is that there exist a marked discrepancy between tail and central dependence structures when the innovations { ε t } of AR(1) process are non-Gaussian. From Figure 2, it can also be evidenced that AR process with Gaussian innovations is time reversible, while the case of t 1 distributed innovations is time reversible only for extreme quantiles. Above results indicate that the imaginary parts of CSDK are not zero suggesting time-irreversibility.