| multIndepTest {copula} | R Documentation |
Analog of the independence test based on the empirical copula process
proposed by Christian Genest and Bruno Rémillard (see
indepTest) for random vectors. The main
difference comes from the fact that critical values and p-values are
obtainted through the bootstrap/permutation methodology, since, here,
test statistics are not distribution-free.
multIndepTest(x, d, m=length(d), N=1000, alpha=0.05)
x |
Data frame or data matrix containing realizations (one per line) of the random vectors whose independence is to be tested. |
d |
Dimensions of the random vectors whose realizations are given
in x. It is required that sum(d)=ncol(x). |
m |
Maximum cardinality of the subsets of random vectors for
which a test statistic is to be computed. It makes sense to consider
m << p especially when p is large. |
N |
Number of bootstrap/permutation samples. |
alpha |
Significance level used in the computation of the critical values for the test statistics. |
See the references below for more details, especially the last one.
The function "multIndepTest" 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 independence empirical
copula process - see In in the last reference) 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.
I. Kojadinovic and M. Holmes (2008), Tests of independence among continuous random vectors based on Cramér-von Mises functionals of the empirical copula process. submitted.
indepTest,
serialIndepTest,
multSerialIndepTest,
dependogram.
## Consider the following example taken from
## Kojadinovic and Holmes (2008):
n <- 100
## Generate data
y <- matrix(rnorm(6*n),n,6)
y[,1] <- y[,2]/2 + sqrt(3)/2*y[,1]
y[,3] <- y[,4]/2 + sqrt(3)/2*y[,3]
y[,5] <- y[,6]/2 + sqrt(3)/2*y[,5]
nc <- normalCopula(0.3,dim=3)
x <- cbind(y,rcopula(nc,n),rcopula(nc,n))
x[,1] <- abs(x[,1]) * sign(x[,3] * x[,5])
x[,2] <- abs(x[,2]) * sign(x[,3] * x[,5])
x[,7] <- x[,7] + x[,10]
x[,8] <- x[,8] + x[,11]
x[,9] <- x[,9] + x[,12]
## Dimensions of the random vectors
d <- c(2,2,2,3,3)
## Run the test
test <- multIndepTest(x,d)
test
## Display the dependogram
dependogram(test)