In an earlier paper,

In the last decades the bootstrap methodology has become more and more
widespread in different areas of statistical applications. See, e.g.,

In Sect. 2, we first briefly review the importance of stationarity of time
series. In the bivariate case, the
vector autoregression (VAR) process
is one of the most important models, becoming popular first in the area of
econometrics

In Sect. 3, we introduce the concept of copulas, the most convenient objects
for analysing the dependence structures among variables. Their history goes
back as far as

Section 4 is devoted to the bootstrap resampling method, including the block
bootstrap approach, which is suitable for the case of serially dependent
observations. Here we introduce a generalisation, which helps overcoming the
problem that originally the block size was supposed to be a natural number.
In our approach the block size is a random variable with arbitrary positive
real-valued expectation greater than 1, and it contains the original block
bootstrap as a special case. Due to this small variance in the sample size,
it overcomes the problem of extensive random error in the case of the
stationary bootstrap (

Section 5 shows the results of our simulations regarding the properties of the proposed homogeneity test. It turned out that it is consistent and it has reasonable power for relatively small sample sizes. We have also investigated the effect of the block size for the properties of the test.

In Sect. 6 we apply our approach to the gridded temperature data base of
E-OBS, which is a product of the EU-FP6 project ENSEMBLES

The conclusion summarises our findings and gives some interesting open questions.

We call the

One of the most frequently applied time series models are the
autoregressive (AR) processes and their multidimensional counterparts,
vector autoregressive (VAR) models. In the following, we define the VAR(

Any VAR(

For the remainder of this section we assume that

In the applications we will use the covariance matrix of the sample mean. The
following asymptotic result will be crucial in our investigations: if

It is important to check if the chosen time series model is adequate. If the
model fits well, the fitted residuals should behave as a realisation of a
white noise process. The hard part is to check whether the residuals are
independent, thus there is no serial dependence among them. There are several
methods for verifying this property, the most standard is the Ljung–Box test,
which tests whether a specified group (usually the first 10–20 lags) of
autocorrelations is different from zero. Another often applied serial
correlation test is the Breusch–Godfrey test. A more recent multidimensional
approach was published in

For further details about time series analysis, see, for example,

Bivariate data and the corresponding pseudo-observations.

Let

In this paper we focus on testing the homogeneity of copulas, motivated by the question of whether climate change also has an effect on the dependence between pairs of temperature observations. If this change is indeed observable, then it may have a substantial effect on the spatial structure of temperature anomalies, worth further meteorological investigations. So we do not have to go into the parametric inference, as we are just interested in the homogeneity analysis.

Let us suppose we have two independent samples of

The proposed tests for checking the homogeneity of two samples are based on
functionals of the empirical process:

The bootstrap is a usually computer-intensive, resampling method for
estimating the distribution of a statistic of interest. The concept of the
bootstrap was introduced in the classical article by Bradley Efron

Let

In our case we are interested in the effect of serial dependence on the homogeneity tests and on modelling in general, for example on the covariance matrix of our estimators. If the data are dependent then the estimates based on i.i.d. bootstrap methods may not be consistent.

In the presence of serial dependence, one of the most commonly used methods
is the block bootstrap, see

Block length plays an important role in the process, and it is not trivial to
determine its optimal value.

We used a similar approach in our previous paper

In

In the literature, simulations are naturally based on integer block sizes.
But using the block length of Eq. (

In case of

In the same way as the circular block bootstrap sample, our generalised
bootstrap sample is not a stationary process, conditional on the original
sample. It is an important theoretical result of

The covariance matrix

We have to mention that the Politis and White algorithm actually gives a
real number and not an integer as the optimal block size – this could
be used without any rounding by our proposed method. In their original paper

Note as well that the type of block length that would be best for the block
bootstrap method depends on the inference problem (e.g. variance estimation
or testing), as described in

As the limit distribution of the statistic

Compute the statistic

Generate

Compute the statistics

Compute the

Guaranteeing the homogeneity of copulas in the context of bootstrap
approaches is still an unsolved problem in general. Traditional bootstrap
approaches have been claimed to be inconsistent for Cramér–von Mises statistics

In this section we present some properties of the copula homogeneity test
obtained via simulations, strongly focusing on a specific VAR(

By using the bootstrap methodology, we can investigate the significance level
of the homogeneity test. Our simulations indicate that the test is consistent
for each block size and each relevant time series model. However, we find,
that for VAR(

Simulated quantiles of the homogeneity test statistic for the VAR(1)
process approximating the temperature data close to Budapest and Apatovac with
sample size

Now we estimate the power of the proposed homogeneity test. We take samples
from the VAR(

The block size does not have a big effect on the power. We have to mention that
although the numbers manifest consistency, the last digit of the powers in
the table may not be accurate. We took 100 samples with 100 bootstrap
replicates (let us call these

The power of the homogeneity test (%). Null hypothesis: the
VAR(

As we shall see in the next section, for the data pairs Budapest and Apatovac,
the block size of 8.71 is going to be a good choice. Table

The power of the homogeneity test (%). Null hypothesis: the VAR(1)
process of Budapest and Apatovac (

We also simulated the test statistic for other models: if the reference

i.i.d. bivariate normal distributed;

an MA(4) process;

a stationary GARCH(1,1) process.

The power of the homogeneity test (%). Null hypothesis: GARCH(1,1)
process with different block sizes, with sample size

The observations comprise 63 years of daily temperature data of the European
Climate Assessment (E-OBS;

The quality of the data has been evaluated, e.g., in

The map of the Carpathian Basin with the used grid points.

As we intend to use models, suitable for stationary data, first the
stationarity had to be ensured. We have first subtracted the smoothed daily
averages from the observations. The smoothing was made by loess regression;
Fig.

Results of tests checking for serial dependence between the
estimated residuals (

In order to reduce the strong serial dependence, we have finally computed the
10-day averages of the

Optimal block length for the first half of the samples for the five selected pairs of grid points.

The pseudo-observations of Budapest and Apatovac for the first and the second half of the 10-day averages of the standardised observations.

In the next step, we examined the fixed grid point near Budapest paired with
other grid points of the database. Using the Akaike information criterion, we
chose the orders of the most appropriate vector autoregression to model our
data pairs. Despite the adjusted

Our main goal is to detect if there is a significant change in the dependence
structure of the data. We separated the pairs of points into two parts – the
first part corresponds to the first 31.5 years' observations and the second
part to the second 31.5 years' observations. For five selected pairs of grid
points, we wanted to test the null hypothesis that the copula of first half
of the sample is equal to the copula of the second half of the sample. We
have tested the independence by the Cramér–von Mises type test of

Table

Generally, we noticed, that as the block sizes tend to be smaller, the trace
function is closer to be monotonic. This phenomenon can be explained by the
expansion of

The last step was conducting the copula homogeneity test described in
Sect.

In this paper, we have used the bootstrap for determining the

We can summarise our findings as follows.

First, we proposed a simple generalisation of the block bootstrap methodology, which fits naturally to the existing algorithms, and which helps to overcome the problem of discreteness in the usual block size. The proposed generalised block bootstrap method can easily be applied to any other problem, where the block size plays an important role, as all block length determining algorithms give a real number as estimated block size.

Second, we have found some significant changes in the dependence structure between the standardised temperature values of pairs of stations within the Carpathian Basin. The direction of this change may be worth further investigation, as this may lead to a better understanding of the recent changes in our climate.

It is an interesting open question, to which models and inference problems the proposed block size determining method – based on functions of the variances – can be successfully applied. We have checked by simulation that the method can be applied to the specific VAR models, described in our article. For non-linear time series we might need more observations to get a fit, which is similarly reliable. As a general comment on the use of bootstrap methods, we have seen cases when the block size did not play an important role, but in our opinion this is rather an exception than a rule. Choosing a not optimal block size may decrease the accuracy of the applied method somewhat, but not using any type of block bootstrap may distort the results completely as quite a few of the references have already demonstrated.

The E-OBS dataset is regularly refreshed. The most
up-to-date version is 15.0, dated June 2017. However at the time of the
writing of the paper, 12.0 was the most recent, which is still available from
the webpage

The authors declare that they have no conflict of interest.

We acknowledge the E-OBS data set from the EU-FP6 project ENSEMBLES
(