dCovTS (version 1.0)

mADCFtest: Distance Correlation test of independence in multivariate time series

Description

A test of independence based on distance correlation function in multivariate time series proposed by Fokianos and Pitsillou (2016).

Usage

mADCFtest(x, type = c("truncated", "bartlett", "daniell", "QS", "parzen"), p,
          b = 0, parallel = FALSE)

Arguments

x

multivariate time series.

type

character string which indicates the smoothing kernel. Possible choices are 'truncated' (the default), 'bartlett', 'daniell', 'QS', 'parzen'.

p

bandwidth, whose choice is determined by $p=cn^{\lambda}$ for $c > 0$ and $\lambda \in (0,1)$.

b

the number of bootstrap replicates of the test statistic. It is a positive integer. If b=0 (the default), then no p-value is returned.

parallel

logical value. By default, parallel=FALSE. If parallel=TRUE, bootstrap computation is distributed to multiple cores, which typically is the maximum number of available CPUs and is detecting directly from the function.

Value

• An object of class htest which is a list containing:
• method
  description of test.
• statistic
  the observed value of the test statistic.
• replicates
  bootstrap replicates of the test statistic (if $b=0$ then replicates=NULL).
• p.value
  p-value of the test (if $b=0$ then p.value=NA).
• data.name
  description of data (data name, kernel type, type, bandwidth, p, and the number of bootstrap replicates, b).

Details

mADCFtest performs a test of multivariate independence. In particular, the function computes a test statistic for testing whether the data are i.i.d. The p-value of the test is obtained via the independent wild bootstrap (Shao, 2010; Leucht and Neumann, 2013), with b replicates. The observed statistic is given by $$\sum_{j=1}^{n-1}(n-j)k^2(j/p)\mbox{tr}{\hat{V}^{*}(j)\hat{D}^{-1}\hat{V}(j)\hat{D}^{-1}}$$ where $\hat{D}^{-1}=diag{\hat{V}_{11}(0), \dots, \hat{V}_{dd}(0)}$ with $d$ indicating the dimension of the multivariate time series and $\hat{V}_{rm}(0)$ is obtained from the elements of the corresponding matrix mADCV. $\hat{V}^{*}(\cdot)$ denotes the complex conjugate matrix of $\hat{V}(\cdot)$ obtained from mADCV, and $\mbox{tr}{A}$ denotes the trace of a matrix $A$. $k(\cdot)$ is a kernel function computed by kernelFun and p is a bandwidth or lag order whose choice is further discussed in Fokianos and Pitsillou (2016). Under the null hypothesis of independence and some further assumptions about the kernel function $k(\cdot)$, the standardized version of the test statistic follows $N(0,1)$ asymptotically and it is consistent. More details of the asymptotic properties of the statistic can be found in Fokianos and Pitsillou (2016). mADCVtest performs the same test based on the distance covariance matrix mADCV.

References

Fokianos K. and M. Pitsillou (2016). On multivariate auto-distance covariance and correlation functions. Submitted for publication. Leucht, A. and M. H. Neumann (2013). Dependent wild bootstrap for degenerate U- and V- statistics. Journal of Multivariate Analysis $\textbf{117}$, 257-280, http://dx.doi.org/10.1016/j.jmva.2013.03.003. Shao, X. (2010). The dependent wild bootstrap. Journal of the American Statistical Association $\textbf{105}$, 218-235, http://dx.doi.org/10.1198/jasa.2009.tm08744.

See Also

mADCF, mADCV, mADCVtest

Examples

x<-MASS::mvrnorm(300,rep(0,2),diag(2))
n <- length(x)
c <- 3
lambda <- 0.1
p <- ceiling(c*n^lambda)
mT=mADCFtest(x,type="tr",p=p,b=499,parallel=TRUE)
mF=mADCFtest(x,type="tr",p=p,b=499,parallel=FALSE)