| serialIndepTest {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 indepTest, 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.
serialIndepTestSim(n, lag.max, m=lag.max+1, N=1000) serialIndepTest(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 serialIndepTestDist as returned by the
function serialIndepTestSim. 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 "serialIndepTestSim" returns an object of class
"serialIndepTestDist" 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 "serialIndepTest" returns an object of class
"indepTest" 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).
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.
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 (2006). Local efficiency of a Cramer-von Mises test of independence, Journal of Multivariate Analysis, 97:274–294.
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.
indepTest,
multIndepTest,
multSerialIndepTest,
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 <- serialIndepTestSim(100,3)
## The next step consists in performing the test itself:
test <- serialIndepTest(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.