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)