Test proposed by Zhou et al. (2017) — glhtbf

Zhou et al. (2017)'s test for general linear hypothesis testing (GLHT) problem for high-dimensional data under heteroscedasticity.

Usage

glhtbf_zgz2017(Y,G,n,p)

Arguments

Y: A list of $k$ data matrices. The $i$th element represents the data matrix ($p\times n_i$) from the $i$th population with each column representing a $p$-dimensional observation.
G: A known full-rank coefficient matrix ($q\times k$) with $\operatorname{rank}(\boldsymbol{G})< k$.
n: A vector of $k$ sample sizes. The $i$th element represents the sample size of group $i$, $n_i$.
p: The dimension of data.

Value

A (list) object of S3 class htest containing the following elements:

statistic: the test statistic proposed by Zhou et al. (2017).
p.value: the $p$-value of the test proposed by Zhou et al. (2017).

Details

Suppose we have the following $k$ independent high-dimensional samples: $$ \boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma}_i,i=1,\ldots,k. $$ It is of interest to test the following GLHT problem: $$H_0: \boldsymbol{G M}=\boldsymbol{0}, \quad \text { vs. } H_1: \boldsymbol{G M} \neq \boldsymbol{0},$$ where $\boldsymbol{M}=(\boldsymbol{\mu}_1,\ldots,\boldsymbol{\mu}_k)^\top$ is a $k\times p$ matrix collecting $k$ mean vectors and $\boldsymbol{G}:q\times k$ is a known full-rank coefficient matrix with $\operatorname{rank}(\boldsymbol{G})<k$.

Let $\bar{\boldsymbol{y}}_{i},i=1,\ldots,k$ be the sample mean vectors and $\hat{\boldsymbol{\Sigma}}_i,i=1,\ldots,k$ be the sample covariance matrices.

Zhou et al. (2017) proposed the following U-statistic based test statistic: $$ T_{ZGZ}=\|\boldsymbol{C \hat{\mu}}\|^2-\sum_{i=1}^k h_{ii}\operatorname{tr}(\hat{\boldsymbol{\Sigma}}_i)/n_i, $$ where $\boldsymbol{C}=[(\boldsymbol{G D G}^\top)^{-1/2}\boldsymbol{G}]\otimes\boldsymbol{I}_p$, $\boldsymbol{D}=\operatorname{diag}(1/n_1,\ldots,1/n_k)$, and $h_{ij}$ is the $(i,j)$th entry of the $k\times k$ matrix $\boldsymbol{H}=\boldsymbol{G}^\top(\boldsymbol{G}\boldsymbol{D}\boldsymbol{G}^\top)^{-1}\boldsymbol{G}$.

They showed that under the null hypothesis, $T_{ZGZ}$ is asymptotically normally distributed.

References

Zhou B, Guo J, Zhang J (2017). “High-dimensional general linear hypothesis testing under heteroscedasticity.” Journal of Statistical Planning and Inference, 188, 36--54. doi:10.1016/j.jspi.2017.03.005 .

Examples

set.seed(1234)
k <- 3
p <- 50
n <- c(25, 30, 40)
rho <- 0.1
M <- matrix(rep(0, k * p), nrow = k, ncol = p)
avec <- seq(1, k)
Y <- list()
for (g in 1:k) {
  a <- avec[g]
  y <- (-2 * sqrt(a * (1 - rho)) + sqrt(4 * a * (1 - rho) + 4 * p * a * rho)) / (2 * p)
  x <- y + sqrt(a * (1 - rho))
  Gamma <- matrix(rep(y, p * p), nrow = p)
  diag(Gamma) <- rep(x, p)
  Z <- matrix(rnorm(n[g] * p, mean = 0, sd = 1), p, n[g])
  Y[[g]] <- Gamma %*% Z + t(t(M[g, ])) %*% (rep(1, n[g]))
}
G <- cbind(diag(k - 1), rep(-1, k - 1))
glhtbf_zgz2017(Y, G, n, p)
#> 
#> 
#> 
#> data:  
#> statistic = 3.3665, p-value = 0.4587
#>