empcops.test {copula} | R Documentation |
Serial independence test based on the empirical
copula process as proposed in Ghoudi et al. (2001) and Genest and
Rémillard (2004). The test, which is the serial analog of
empcopu.test
,
can be seen as composed of three steps: (i) a simulation step, which consists in simulating the
distribution of the test statistics under serial independence for the sample
size under consideration; (ii) the test itself, which consists in
computing the approximate p-values of the test statistics with respect
to the empirical distributions obtained in step (i); and (iii) the
display of a graphic, called a dependogram, enabling to
understand the type of departure from serial independence, if any. More details can
be found in the articles cited in the reference section.
empcops.simulate(n, lag.max, m=lag.max+1, N=1000) empcops.test(x, d, alpha=0.05)
n |
Length of the time series when simulating the distribution of the test statistics under serial independence. |
lag.max |
Maximum lag. |
m |
Maximum cardinality of the subsets of 'lags' for which a test statistic
is to be computed. It makes sense to consider m << lag.max+1 especially when
lag.max is large. |
N |
Number of repetitions when simulating under serial independence. |
x |
Numeric vector containing the time series whose serial independence is to be tested. |
d |
Object of class empcops.distribution as returned by
the function empcops.simulate . It can be regarded as the empirical distribution
of the test statistics under serial independence. |
alpha |
Significance level used in the computation of the critical values for the test statistics. |
See the references below for more details, especially the third and fourth ones.
The function empcops.simulate
returns an object of class
empcops.distribution
whose attributes are: sample.size
,
lag.max
, max.card.subsets
,
number.repetitons
, subsets
(list of the subsets for
which test statistics have been computed), subsets.binary
(subsets in binary 'integer' notation),
dist.statistics.independence
(a N
line matrix containing the values of the test statistics for each subset and each repetition)
and dist.global.statistic.independence
(a vector a length N
containing
the values of the serial version of the global Cramér-von Mises test statistic for each repetition
- see last reference p 175).
The function empcops.test
returns an object of class
empcop.test
whose attributes are: subsets
,
statistics
, critical.values
, pvalues
,
fisher.pvalue
(a p-value resulting from a combination à la
Fisher of the subset statistic p-values), tippett.pvalue
(a p-value
resulting from a combination à la Tippett of the subset
statistic p-values),
alpha
(global significance level of the test), beta
(1 - beta
is the significance level per statistic),
global.statistic
(value of the global Cramér-von Mises
statistic derived directly from the serial independence
empirical copula process - see last reference p 175) and
global.statistic.pvalue
(corresponding p-value).
Ivan Kojadinovic, ivan@stat.auckland.ac.nz
P. Deheuvels (1979), La fonction de dépendance empirique et ses propriétés: un test non paramétrique d'indépendance, Acad. Roy. Belg. Bull. Cl. Sci. 5th Ser. 65, 274-292.
P. Deheuvels (1981), A non parametric test for independence, Publ. Inst. Statist. Univ. Paris 26, 29-50.
K. Ghoudi, R. Kulperger, and B. Rémillard (2001), A nonparametric test of serial independence for times series and residuals, Journal of Multivariate Analysis, 79:191-218.
C. Genest and B. Rémillard (2004), Tests of independence and randomness based on the empirical copula process, Test 13, 335-369.
C. Genest, J.-F. Quessy and B. Rémillard (2007), Asymptotic local efficiency of Cramér-von Mises tests for multivariate independence, The Annals of Statistics 35, 166-191.
empcopu.test
,empcopm.test
,empcopsm.test
,dependogram
## AR 1 process ar <- numeric(200) ar[1] <- rnorm(1) for (i in 2:200) ar[i] <- 0.5 * ar[i-1] + rnorm(1) x <- ar[101:200] ## In order to test for serial independence, the first step consists ## in simulating the distribution of the test statistics under ## serial independence for the same sample size, i.e. n=100. ## As we are going to consider lags up to 3, i.e., subsets of ## {1,...,4} whose cardinality is between 2 and 4 containing {1}, ## we set lag.max=3. This may take a while... d <- empcops.simulate(100,3) ## The next step consists in performing the test itself: test <- empcops.test(x,d) ## Let us see the results: test ## Display the dependogram: dependogram(test) ## NB: In order to save d for future use, the save function can be used.