Tests Based on the GMD
We consider the testing problem
Our test statistic is given by
The null hypothesis is rejected if T„ < cj,_{ha}, where the critical value q,_{ha} is determined such that Рн_{0}(Ти < = «•
We will analyze the behavior of the test for several possible estimators of 7. It is assumed that the mean is known. Let
Assuming {X,} is stationary, it can be shown that 71,„ is a consistent and asymptotically normally distributed estimator of 7 (see, e.g., Brockwell and Davis, 1991, p. 227).
Further possible estimators of 7 are, e.g., the absolute mean difference
and the median absolute deviation
In both cases, it is again assumed that the mean p is known.
Of course, there are many other possibilities to estimate the standard deviation, but in this paper we will focus on the above estimators which seem to be the most popular ones.
Theorem 9.2. Suppose that the assumptions of Theorem 2.1 are satisfied. Let X = (Xj,... ,X„Y follow a multivariate ellipticalh/ contoured distribution (see, e.g., Fang 1990; Gupta, 2013), briefly X ~ £C„(p, X,
Proof: Let i = 1. The stochastic representation of a multivariate ellipti cally contoured distribution says that (cf. Fang and Zhang, 1990, p. 65)
X = ц + RA'U where A'A = E, R is a nonnegative random variable, U is uniformly distributed on the unit sphere S„ = {u e IR" : u'u = 1}, and R and U are independently distributed. Let A' = (д'),_{=1}_ Consequently,
This shows that the distribution of T„ does not depend on ц and R. It is distribution invariant within the class of multivariate elliptically contoured distributions. The proof for the other test statistics could be given in the same way. ■
This is a very important and extremely useful property of the above test statistics, since the critical values have to be determined only for one single multivariate elliptically contoured distribution. Note that the multivariate normal and the multivariate tdistribution belong to this distribution family (cf. Fang and Zhang, 1990;Gupta et al., 2013).
Following the above argumentations, the null hypothesis will be rejected if Tf is smaller than the critical value qff.
Since the exact distribution of the test statistics is unknown, we will determine the critical values and the power function within a simulation study. However, statements about the asymptotic behavior of the distribution of our test statistics can be obtained by making use of results for Ustatistics. The asymptotic distribution of Ustatistics has been studied for various mixing conditions (e.g., Dehling and Wendler, 2010), i.e., for special types of correlation structures. Theorem 1.8 of Dehling and Wendler (2010) extends the central limit theorem to strongly mixing data. Since the GMD is a special case of a Ustatistic, and if the estimator of 7 is consistent, then our test statistic is asymptotically normal and its asymptotic variance is given in Theorem 1.8 of Dehling and Wendler (2010).
Note that in our study, we assume that ц is known. Of course, this is a restriction in many cases. However, we consider statistical process control as one of the major potential fields of application and in this area ц presents the target value which is usually known or at least can be well estimated using previous observations. Nevertheless, we analyzed the power functions of the tests if in the above estimators for the variance the expectation had been replaced by the mean. It turned out that the corresponding tests get very poor. One explanation for this behavior is that the sample variance is biased for finite samples and that it underestimates the true value (cf. Perci val, 1993; Fuller, 2009). One possibility to overcome this problem seems to be the utilization of nearly unbiased estimators for the variance of a time series with unknown mean as proposed by Vogelsang and Yang (2016). We will not deepen this discussion here and refer to a forthcoming paper.
Analysis of the Power Function of the Test Based on Tn(1)
In this section, we focus on the test statistic . In order to examine its power function, we will consider its behavior for two possible correlation structures. We assume that the data follow a multivariate elliptically contoured distribution. The test based on the GMD is compared with the test of Box and Ljung (see Brockwell and Davis, 1991), which is mostly applied to check for the presence of autocorrelations. Note that the distribution of the test statistic of the LjungBox test is also invariant within the family of multivariate elliptically contoured distributions. The number of lags of the LjungBox test was chosen to be equal to m = [и/10]. In order to have a fair comparison, the critical values for both tests are chosen such that the type I error is equal to a = 5%. For the LjungBox test, the null hypothesis will be rejected if the corresponding test statistic takes values larger than BLi__{q}. 10^{6} independent samples of size n are generated and the critical values are estimated by the corresponding «quantile, and the power functions are estimated by the relative amount of rejections of the null hypothesis. In Table 9.1 the simulated critical values of the test based on and the LjungBox test are given. The values are rounded to three decimal places.
Note that due to the invariance property, it is sufficient to consider the power functions of the GMD test and LjungBox test assuming a multivariate normal distribution. In our simulation study, we always assume that 7 = 1.
We consider two types of correlation structures. The first correlation structure is given by
where Л e {0,0.1,0.2,..., 0.9}. We refer to this structure as case I. In this case, the correlation decreases linearly with the lag, and thus a strong correlation structure is present.
TABLE 9.1
Simulated Critical Values of the Test Statistic T;/^{1} and of the LjungBox Test for Multivariate Elliptically Contoured Distributions as a Function of a and the Sample Size n
Sample Size, и 

a 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
20 
25 
30 
50 

Simulated 
<7^{(1)} Vo.oi 
0.532 
0.626 
0.696 
0.748 
0.79 
0.823 
0.85 
0.873 
0.892 
0.908 
0.923 
0.971 
1.001 
1.02 
1.059 
critical values of 

0.779 
0.847 
0.892 
0.924 
0.949 
0.968 
0.982 
0.995 
1.005 
1.013 
1.021 
1.046 
1.06 
1.07 
1.09 
4'of LjungBox 
lio.i 
0.92 
0.957 
0.987 
1.007 
1.021 
1.032 
1.041 
1.049 
1.055 
1.06 
1.064 
1.078 
1.087 
1.092 
1.103 
test 
ИО.Ч5 
4.11 
3.969 
3.987 
3.975 
3.966 
3.96 
3.962 
3.951 
3.943 
3.951 
3.948 
5.822 
5.868 
7.885 
11.367 
The second correlation function (case II) is given by = AMI, i,j e {1,...,« i ф j} with A € {0,0.1,.. .,0.9}. Here, we observe an exponential decrease, and thus a weaker correlation than in case I.
The power functions of the tests are given in Figures 9.1a and 9.1b. Figure 9.1a shows that for case I the proposed test turns out to be much better than the LjungBox test for all sample sizes and all values of A considered in our study. The results of LjungBox are really poor. It can also be seen that the power function of the test increases with the sample size n.
In Figure 9.1b a different behavior can be observed than in Figure 9.1a. The test based on dominates the LjungBox test only for smaller values of n and A. Its performance does not change if n increases from 10 to 15 while the power function of LjungBox rapidly increases with the sample size.
Our comparison study has shown that the performance of the considered tests depends on the correlation structure, i.e., the parameter constellation under the alternative hypothesis. However, the proposed test turns out to be better than the LjungBox test for small samples sizes (e.g., n < 10). Such small sample sizes can be frequently observed in statistical process control. This is the reason why we want to focus on possible applications in statistical process control. We will analyze a possible application in more detail in Section 9.3.
FIGURE 9.1
Comparison of the empirical power functions of the test based on (without bullets ) with the LjungBox test (with bullets) for different sample sizes: (a) — correlation structure I, (b) — correlation structure II
TABLE 9.2
Critical Values of the Test Statistics T;,^{2}' and T,^{(},^{3)} for Different Sample Sizes n
Sample Size, it 

5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 

Empirical 
0.905 
0.995 
1.055 
1.102 
1.135 
1.161 
1.184 
1.202 
1.216 
1.229 
1.24 

0.05quantiles 
0.877 
1.005 
1.035 
1.101 
1.106 
1.153 
1.151 
1.189 
1.185 
1.216 
1.212 
Comparison of Several Tests Based on the GMD
In Section 9.2.3 the main focus was on the analysis of the new test based on Tj,^{1 1} and its comparison with the LjungBox test. Here, we will discuss further tests based on the GMD. They differ from the one considered in the last section by using other estimators of the standard deviation. We restrict ourselves to T,^{(},^{2)} and T,^{(},^{3)} (cf. Section 9.2.2).
In our comparison study we use the same framework as in the previous section. In particular, we restrict ourselves to the two correlation structures described above. The critical values and the power functions are obtained using 10^{6} simulations as described in Section 9.2.3. The critical values of the tests Tjr, i = 2,3 are given in Table 9.2 and the corresponding power functions in Figures 9.2 and 9.3. In Figure 9.2 the results for the first correlation structure, I, are plotted while in Figure 9.3 the results for the second one, i.e., II, are shown.
A comparison of Figures 9.1 and 9.2 shows that for correlation structure I the test using T*,^{11} provides the best power function followed by T,^{(},^{2)}. The test based on T,*,^{3)} behaves worse but still better than the LjungBox test. The performance of the tests increases with the sample size.
For correlation structure II, we observe a similar result as in Figure 9.1b. The new tests only dominate the LjungBox test for small values of n and A. Within the new tests Tj,^{1} * and Ту have similar performance and the test using tI,^{3>} is the worst one. For correlation structure II the LjungBox test dominates the others if n and A are sufficiently large.
The great advantage of the proposed tests lies in using them for the case of small sample sizes. They show quite good performance and outstrip the LjungBox test. Moreover, they have the nice property that the distributions of the corresponding test statistics are invariant within the family of multivariate elliptically contoured distributions. If the mean is known, we recommend using the test based on T„ .
A drawback of the proposed procedure is that it is only able to detect an increase in the correlation and not a decrease. This is, of course, a disadvantage with respect to the LjungBox test. However, in many practical situations as, e.g., in engineering, we know that the data are not negatively correlated
FIGURE 9.2
Comparison of the empirical power functions of the tests based on (a) and Tf^{1} (b) (without bullets) with the LjungBox test (with bullets) for correlation structure I
FIGURE 9.3
Comparison of the empirical power functions of the tests based on (a) and Tf^{1} (b) (without bullets) with the LjungBox test (with bullets) for correlation structure II and thus, the main question is whether the data are uncorrelated or positively correlated.