copula (version 0.6-1)

empcop.rv.test: Independence test among continuous random vectors based on the empirical copula process

Description

Analog of the independence test based on the empirical copula process proposed by Christian Genest and Bruno R�millard (see empcop.test) for random vectors. The main difference comes from the fact that critical values and p-values are obtainted through the bootstrap methodology, since, here, test statistics are not distribution-free.

Usage

empcop.rv.test(x, d, m=length(d), N=1000, alpha=0.05)

Arguments

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 samples.

alpha

Significance level used in the computation of the critical values for the test statistics.

Value

• The function empcop.rv.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 � la Fisher of the subset statistic p-values), tippett.pvalue (a p-value resulting from a combination � 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 Cram�r-von Mises statistic derived directly from the independence empirical copula process - see In in the last reference) and global.statistic.pvalue (corresponding p-value).

encoding

latin1

Details

See the references below for more details, especially the last one.

References

P. Deheuvels (1979), La fonction de d�pendance empirique et ses propri�t�s: un test non param�trique d'ind�pendance, 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. R�millard (2004), Tests of independence and randomness based on the empirical copula process, Test 13, 335-369. C. Genest, J.-F. Quessy and B. R�millard (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. R�millard (2007), Asymptotic local efficiency of Cram�r-von Mises tests for multivariate independence, The Annals of Statistics 35, 166-191. I. Kojadinovic and M. Holmes (2007), Tests of independence among continuous random vectors based on Cram�r-von Mises functionals of the empirical copula process, submitted.

See Also

empcop.test

Examples

## 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 <- empcop.rv.test(x,d,N=2000)
test

## Display the dependogram
dependogram(test)