jdcov.test: Test for joint independence based on joint distance...
In shubhadeep4/jdcov: Tests for joint independence via joint distance covariance

Description Usage Arguments Value References Examples

Performs a bootstrap based test for joint independence among d(>=2) random vectors X_1, .. , X_d of (arbitrary) dimensions p_1, .. , p_d, described in Section 4 in Chakraborty and Zhang (2019). The test is based on the joint distance covariance (JdCov) among X_1, .. , X_d, which is shown to completely characterize joint independence among the random vectors. The null hypothesis (H_0) is that all the random vectors are jointly independent.

1	jdcov.test(x, cc = 1, B = 100, stat.type = "U", alpha = 0.05)

`x`	a list of d (>=2) elements, the i^th element being a numeric n*p_i data matrix for the random vector X_i, i = 1,..,d. n denotes the sample size and p_i denotes the dimension of the i^th random vector X_i.
`cc`	a numeric value specifying the choice of the tuning parameter c, default is 1.
`B`	an integer value specifying the number of bootstrap replicates to be considered, default is 100.
`stat.type`	a character string specifying the type of the estimator of joint distance covariance to be computed. The available options are : "V" (the V-statistic type estimator), "U" (the U-statistic type estimator), "US" (the scale invariant U-statistic type estimator), "Rank V" (the rank based V-statistic type estimator) and "Rank U" (the rank based U-statistic type estimator). Default is "U".
`alpha`	a numeric value specifying the level of the test, default is 0.05.

A list containing the following components :

statistic : the observed value of the test statistic

crit.value : the critical value of the bootstrap-based hypothesis test. The null hypothesis (H_0: joint independence) is rejected if the observed value of the statistic is greater than crit.value

p.value : the p-value of the bootstrap-based hypothesis test. The null hypothesis (H_0: joint independence) is rejected if the p.value is smaller than alpha.

Chakraborty, S. and Zhang, X. (2019). Distance Metrics for Measuring Joint Dependence with Application to Causal Inference, Journal of the American Statistical Association, DOI: 10.1080/01621459.2018.1513364.

## (X_1, .. , X_d) are generated as follows : X=Z^3, where
## Z~N(0,diag(d)). 

library(mvtnorm)
n=100; d=5
set.seed(10)
z=rmvnorm(n,mean=rep(0,d),sigma=diag(d)) ; x=z^3
X <- lapply(seq_len(ncol(x)), function(i) as.matrix(x[,i]))
jdcov.test(X, cc=1, B=100, stat.type = "U", alpha=0.05) 


## X, Y and Z are p-dimensional random vectors, where X, Y  
## are i.i.d N(0, diag(p)), Z_1 = sign(X_1 * Y_1) * W and 
## Z_{2:p} ~ N(0, diag(p-1)), W ~ exponential(mean=sqrt(2)). 

library(mvtnorm)
n=100 ; d=3; p=5
x=list() ; x[[1]]=x[[2]]=x[[3]]=matrix(0,n,p)
set.seed(1)
x[[1]]=rmvnorm(n,rep(0,p),diag(p)) 
set.seed(2)
x[[2]]=rmvnorm(n,rep(0,p),diag(p))
set.seed(3)
W=rexp(n,1/sqrt(2))
x[[3]][,1]=(sign(x[[1]][,1] * x[[2]][,1])) * W
set.seed(4)
x[[3]][,2:p]=rmvnorm(n,rep(0,(p-1)),diag(p-1))
jdcov.test(x, cc=1, B=100, stat.type = "U", alpha=0.05)