Independence Test Among Continuous Random Vectors Based on the Empirical Copula Process
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
obtained through the bootstrap/permutation methodology, since, here,
test statistics are not distribution-free.
multIndepTest(x, d, m = length(d), N = 1000, alpha = 0.05,
verbose = interactive())x |
data frame ( |
d |
dimensions of the random vectors whose realizations are given
in |
m |
maximum cardinality of the subsets of random vectors for
which a test statistic is to be computed. It makes sense to consider
|
N |
number of bootstrap/permutation samples. |
alpha |
significance level used in the computation of the critical values for the test statistics. |
verbose |
a logical specifying if progress
should be displayed via |
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).
The former argument print.every is deprecated and not
supported anymore; use verbose instead.
Deheuvels, P. (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.
Deheuvels, P. (1981), A non parametric test for independence, Publ. Inst. Statist. Univ. Paris. 26, 29–50.
Genest, C. and Rémillard, B. (2004), Tests of independence and randomness based on the empirical copula process. Test 13, 335–369.
Genest, C., Quessy, J.-F., and Rémillard, B. (2006). Local efficiency of a Cramer-von Mises test of independence, Journal of Multivariate Analysis 97, 274–294.
Genest, C., Quessy, J.-F., and Rémillard, B. (2007), Asymptotic local efficiency of Cramér-von Mises tests for multivariate independence. The Annals of Statistics 35, 166–191.
Kojadinovic, I. and Holmes, M. (2009), Tests of independence among continuous random vectors based on Cramér-von Mises functionals of the empirical copula process. Journal of Multivariate Analysis 100, 1137–1154.
## 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(n, nc),rCopula(n, nc)) 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,print=TRUE)
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