copula (version 0.5-8)

empcop.test: Multivariate independence test based on the empirical copula process

Description

Multivariate independence test based on the empirical copula process as proposed by Christian Genest and Bruno R�millard. The test can be seen as composed of three steps: (i) a simulation step, which consists in simulating the distribution of the test statistics under independence for the sample size under consideration; (ii) the test itself, which consists in computing the approximate p-values of the test statistics with respect to the empirical distributions obtained in step (i); and (iii) the display of a graphic, called a dependogram, enabling to understand the type of departure from independence, if any. More details can be found in the articles cited in the reference section.

Usage

empcop.simulate(n, p, m, N = 2000)
empcop.test(x, d, m, alpha = 0.05)
dependogram(test, pvalues = FALSE)

Arguments

n

Sample size when simulating the distribution of the test statistics under independence.

p

Dimension of the data when simulating the distribution of the test statistics under independence.

m

Maximum cardinality of the subsets for which a test statistic is to be computed. It makes sense to consider m << p especially when p is large. In this case, the null hypothesis corresponds to the independence of all sub-ran

N

Number of repetitions when simulating under independence.

x

Data frame or data matrix containing realizations (one per line) of the random vector whose independence is to be tested.

d

Object of class empcop.distribution as returned by the function empcop.simulate. It can be regarded as the empirical distribution of the test statistics under independence.

alpha

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

test

Object of class empcop.test as return by he function empcop.test.

pvalues

Logical indicating whether the dependogram should be drawn from test statistics or the corresponding p-values.

Value

• The function empcop.simulate returns an object of class empcop.distribution whose attributes are: sample.size, data.dimension, max.card.subsets, number.repetitons, subsets (list of the subsets for which test statistics have been computed), subsets.binary (subsets in binary 'integer' notation) and dist.statistics.independence (a N line matrix containing the values of the test statistics for each subset and each repetition).

  The function empcop.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 previous p-values), tippett.pvalue (a p-value resulting from a combination � la Tippett of the previous p-values), alpha (global significance level of the test) and beta (1 - beta is the significance level per statistic).

encoding

latin1

Details

See the references below for more details, especially the third 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 Cramer-von Mises tests for multivariate independence. The Annals of Statistics, 35, in press.

Examples

## Consider the following example taken from
## Genest and Remillard (2004), p 352:

x <- matrix(rnorm(500),100,5)
x[,1] <- abs(x[,1]) * sign(x[,2] * x[,3])
x[,5] <- x[,4]/2 + sqrt(3) * x[,5]/2


## In order to test for independence "within" x, the first step consists
## in simulating the distribution of the test statistics under
## independence for the same sample size and dimension,
## i.e. n=100 and p=5. As we are going to consider all the subsets of
## {1,...,5} whose cardinality is between 2 and 5, we set m=5.
## This may take a while...

d <- empcop.simulate(100,5,5)

## The next step consists in performing the test itself:

test <- empcop.test(x,d,5)

## Let us see the results:

test

## Display the dependogram:

dependogram(test)

## We could have tested for a weaker form of independence, for instance,
## by only computing statistics for subsets whose cardinality is between 2
## and 3:

test <- empcop.test(x,d,3)
test
dependogram(test)

## Consider now the following data:
y <- matrix(runif(500),100,5)
## and perform the test:
test <- empcop.test(y,d,3)
test
dependogram(test)

## NB: In order to save d for future use, the save function can be used.