| 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.