| empcopm.test {copula} | R Documentation |
Analog of the independence test based on the empirical
copula process proposed by Christian Genest and Bruno
Rémillard (see empcopu.test) 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.
empcopm.test(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 empcopm.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 independence empirical copula process - see In in the last reference) 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.
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.
empcopu.test,empcops.test,empcopsm.test,dependogram
## Consider the following example taken from
## Kojadinovic and Holmes (2007):
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 <- empcopm.test(x,d)
test
## Display the dependogram
dependogram(test)