hodcov: High order distance covariance

In shubhadeep4/jdcov: Tests for joint independence via joint distance covariance

Description

Computes the d^th order distance covariance among d(>=2) random vectors X_1, .. , X_d of (arbitrary) dimensions p_1, .. , p_d, defined in Section 2.1 in Chakraborty and Zhang (2019). The population d^th order distance covariance among X_1, .. , X_d is zero if the random vectors are jointly independent (but the converse is not true).

Usage

hodcov(x, type = "U")

Arguments

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.
type	a character, either "U" (computes the U-statistic type estimator of the squared d^th order dCov) or "V" (computes the V-statistic type estimator of the squared d^th order dCov)

Value

Returns the observed value of the U-statistic type or the V-statistic type estimator of the squared d^th order dCov, based on whether the type is "U" or "V".

References

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.

Examples

## (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]))
hodcov(X, type="U")


## 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))
hodcov(x, type="V")

shubhadeep4/jdcov documentation built on June 4, 2019, 3 a.m.