# 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. ```r multIndepTest(x, d, m = length(d), N = 1000, alpha = 0.05, verbose = interactive()) ``` ## Arguments - `x`: data frame (`data.frame`) or 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. - `verbose`: a logical specifying if progress should be displayed via `txtProgressBar`. ## Details See the references below for more details, especially the last one. ## Returns 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. ## References 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 R millard, B. (2004), Tests of independence and randomness based on the empirical copula process. **Test** 13 , 335--369. Genest, C., Quessy, J.-F., and R millard, 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 R millard, 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. ## See Also `indepTest`, `serialIndepTest`, `multSerialIndepTest`, `dependogram`. ## Examples ```r ## 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 (Xtras <- copula:::doExtras()) # or set 'N = ...' manually test <- multIndepTest(x, d, N = if(Xtras) 1000 else 150) test ## Display the dependogram dependogram(test, print=TRUE) ```

multIndepTest function