In this section, we evaluate the performance of the monitoring test proposed in Section 3
through a simulation study. For this task, we use the boundary function in (5
) and employ the stopping rule based on (7
). In this study, we consider the bivariate Gaussian copula with copula parameter
as a true copula model. To see an effect from the copula functions with different functional forms allowing degrees of asymmetry and tail dependence, we also consider the Gumbel copula that is asymmetric and has upper tail dependency as a true model. The copula parameters of the Gumbel model are set to be equal to the value of Kendall’s tau
in Gaussian copula models. For each case, sets of n
= 100, 200 and 300 observations are generated from the copula model with marginal distribution
. The empirical sizes and powers are calculated by the number of rejections of the null hypothesis “
no changes occur in the copula model at
”, out of 1000 repetitions. Here, the predetermined maximal number of observations
are considered as
for empirical size and
for empirical power. In order to examine the power, we consider the following alternative hypotheses. We take into account two elliptical copulas such as the Gaussian and the Student t
and the Frank copula as alternative hypotheses.
A change occurs from the Gaussian copula with to the Gaussian copula with and at .
A change occurs from the Gaussian copula with to the Student t copula with and at .
A change occurs from the Gaussian copula with to the Frank copula with and at .
For the Gumbel copula, we consider Archimedean copulas family for alternative hypotheses such as the Gumbel, Clayton and Frank copulas. For this, we consider the following alternative hypotheses.
A change occurs from the Gumbel copula with to the Gumbel copula with at .
A change occurs from the Gumbel copula with to the Clayton copula with at .
A change occurs from the Gumbel copula with to the Frank copula with at .
In each case, the copula parameters are set to be at the same level in terms of Kendall’s tau in different copula families. To examine the power, many cases of changes in the dependence structure are considered, namely changes of the copula parameter and/or changes of the copula family. For and , we consider the situation involving a change of the copula parameter within a copula family. For , , and , we examined power of the case involving a change of copula parameter and copula family at the same time.
Throughout our simulation study, we only consider the change of copula function and assume that the marginal distributions experience no changes. The empirical sizes are calculated at the nominal levels 0.01, 0.05 and 0.10, and the powers are examined at the nominal level 0.10. The bootstrap method is used for the calculation of the critical value at the nominal level. We perform the bootstrap method discussed in Section 3
and 300, and the constant a
of the boudary function in (5
) is chosen to be 2.
In particular, our test is compared with the monitoring test proposed by Na et al. [21
]. Recall that Na et al. [21
]’s test can be applied to detect a copula parameter change when the copula family does not change. Na et al. [21
] proposed the detector in (2
) based on the difference between estimates of the copula parameter:
is the estimator of the true copula parameter
based on the observation up to time k
is the estimator obtained based on the historical data.
Empirical sizes and powers are presented in Table 1
, Table 2
, Table 3
and Table 4
. The figures in Table 1
, Table 2
, Table 3
and Table 4
while the figures in the parentheses are for
. Table 1
shows that the test procedure has some size distortions when n
is small. However, as n
increases, the empirical size of the test gets very close to the nominal levels in most cases. Size distortions of tests for small sample sizes can be reduced if a smaller a
is chosen. For
, it can be seen that the test also has some size distortions, but the test is generally able to keep their nominal level, especially when
. The result is also same for the other copula models such as t
-copula, Frank copula, and Clayton copula, although not reported here for brevity. Table 2
, Table 3
and Table 4
report the empirical power of
. Table 2
, Table 3
and Table 4
show that our monitoring procedure produces good powers in most cases. It is shown that the powers increase remarkably either as n
increases or the more significant change occurs. Moreover, we can see that when the changes in copula function occur earlier, the powers increase remarkably. It can be seen that the powers in the case that n
is large and
are very close to 1. As pointed out by Lee et al. [22
], our monitoring procedure with boundary function such as (5
) detects early changes more effectively than late changes. Due to the curvature of the component
in the boundary function, the boundary function increases rapidly as the change point moves further away from the point where the monitoring was initiated. This implies that it is more likely to capture small changes early in the sample. Consequently, our test has better power properties for early change points. Similar findings were reported in Na et al. [21
] and Lee et al. [22
]. This result indicates that it is desirable to renew the historical data appropriately to escalate the power when the null hypothesis appears to be true for a certain period time. Note that for alternative hypothesis
involving a change of the copula parameter within a copula family, the monitoring procedure based on
appears to have higher powers than our monitoring test. This result can be explained by the fact that Na et al. [21
]’s monitoring test is designed only to detect parameter changes of copula function. However, even if we consider the alternative hypotheses
that involve a change of copula family, Na et al. [21
]’s test also shows good performance in terms of power. This means that Na et al. [21
]’s test tends to detect a copula parameter change, even though the change actually occurs in copula function. From this aspect, we were motivated to develop the monitoring procedure for detecting a copula function change. In comparing Table 4
against Table 3
, the performance of power appears to be similar. Different functional forms of copula seemed to have no impact on the performance, hence our monitoring test also has good performance in copula models having asymmetry properties or tail dependency. All these results indicate that our test procedure performs adequately to monitor for stability of copula function.
4.2. Real Data Analysis
In this section, we illustrate an example of a real data analysis. We consider bivariate climate data consisting of temperature and precipitation over the contiguous United States. There is a lot of literature studying the association of temperature and precipitation over the United States, and they reported empirical evidence that there is an obvious relationship between two variables (see Zhao and Khalil [35
] and Huang and van Den Dool [36
] and the papers cited therein). Recently, several authors have used a copula based methodology to model the joint distribution of temperature and precipitation (see, e.g., Favre et al. [37
], Shiau et al. [38
], Dupuis [39
] and Schölzel and Friederichs [40
]). However, the precedent studies only focus on the problem as to which copula model best fitted the empirical data. Here, we use the copula functions to model the dependence between temperature and precipitation and attempt to monitor for stability of dependence. Annual mean temperature and annual mean precipitation in summer months (June, July, and August) over the contiguous United States from 1895 to 2015 are used for empirical data. The data can be obtained from NOAA’s National Centers for Environmental Information (NCEI). Figure 1
shows that precipitation and temperature tend to be negatively correlated. It is well known that warmer summers usually result in drier conditions and colder summers are likely to be wetter. For historical data, the data from 1895 to 1975 is used, which has 81 observations. As discussed earlier, since the monitoring test for copula function can be influenced by a change in marginal distribution, the change point tests for marginal distributions are performed in advance of implementing the monitoring test for a copula function change. To this end, we perform the test of Lee et al. [22
] who sequentially monitored marginal distributional changes based on the following test statistic:
is the empirical distribution based on the observation up to time k
is the empirical distribution obtained based on the historical data. By observing new data sequentially, we first conduct the monitoring test for marginal distributional changes. If there are no changes in marginal distributions, we can perform the monitoring test for the copula function change. Since both of the two series detect no evidence of a change in marginal distributions at the nominal level 0.05, we apply monitoring procedure based on the test statistic in (7
) to detect a change of dependence. For this task, we use the boundary function in (5
and perform the bootstrap method in Section 3
. As a result, it appears that the test detects a change in dependence at nominal levels 0.01, 0.05, and 0.10. The location of the stopping time is summarized in Table 5
and Figure 1
illustrates the stopping time in dependence: the solid line corresponds to the end of historical data and the dotted lines identify the detected stopping time.