WN_test: Testing for white noise hypothesis in high dimension

Description

WN_test() is the test proposed in Chang, Yao and Zhou (2017) for the following hypothesis testing problems: $$H_0:\{{\bf x}_t \}_{t=1}^n\mathrm{\ is\ white\ noise\ \ versus\ \ }H_1:\{{\bf x}_t \}_{t=1}^n\mathrm{\ is\ not\ white\ noise.} $$

Usage

WN_test(
  X,
  lag.k = 2,
  B = 2000,
  kernel.type = c("QS", "Par", "Bart"),
  pre = FALSE,
  alpha = 0.05,
  k0 = 5,
  thresh = FALSE,
  tuning.vec = NULL
)

Value

An object of class "hdtstest" is a list containing the following components:

statistic: The value of the test statistic.
p.value: Numerical value which represents the p-value of the test based on the observed data $\{{\bf x}_t\}_{t=1}^n$.
lag.k: The time lag used in function.
method: A character string indicating what method was performed.
kernel.type: A character string indicating what kenel method was performed.

Arguments

X: ${\bf X} = \{{\bf x}_1, \dots , {\bf x}_n \}'$, an $n\times p$ sample matrix, where $n$ is the sample size and $p$ is the dimension of ${\bf x}_t$.
lag.k: Time lag $K$, a positive integer, used to calculate the test statistic [See (4) in Chang, Yao and Zhou (2017)]. Default is lag.k $=2$.
B: Bootstrap times for generating multivariate normal distributed random vectors in calculating the critical value. Default is B $=2000$.
kernel.type: String, an option for choosing the symmetric kernel used in the estimation of long-run covariance matrix, for example, 'QS' (Quadratic spectral kernel), 'Par' (Parzen kernel) and 'Bart' (Bartlett kernel), see Andrews (1991) for more information. Default option iskernel.type = 'QS'.
pre: Logical value which determines whether to performs preprocessing procedure on data matrix X or not, see Remark 1 in Chang, Yao and Zhou (2017) for more information. If TRUE, then the segment procedure will be performed to data X first. The three additional options including thresh, tuning.vec and cv.num are the same as those in PCA_TS.
alpha: The prescribed significance level. Default is 0.05.
k0: A positive integer specified to calculate $\widehat{{\bf W}}_y$. See parameter lag.k in PCA_TS for more information.
thresh: Logical. It determines whether to perform the threshold method to estimate $\widehat{{\bf W}}_y$ or not. See parameter thresh in PCA_TS for more information.
tuning.vec: The value of thresholding tuning parameter $\lambda$. See parameter tuning.vec in PCA_TS for more information.

References

Chang, J., Yao, Q. & Zhou, W. (2017). Testing for high-dimensional white noise using maximum cross-correlations, Biometrika, Vol. 104, pp. 111–127.

Chang, J., Guo, B. & Yao, Q. (2018). Principal component analysis for second-order stationary vector time series, The Annals of Statistics, Vol. 46, pp. 2094–2124.

Cai, T. and Liu, W. (2011). Adaptive thresholding for sparse covariance matrix estimation, Journal of the American Statistical Association, Vol. 106, pp. 672--684.

Examples

Run this code

n <- 200
p <- 10
X <- matrix(rnorm(n*p),n,p)
res <- WN_test(X)
Pvalue <- res$p.value
rej <- res$reject