glhtbf_zzg2022 {HDNRA} | R Documentation |
Zhang et al. (2022)'s test for general linear hypothesis testing (GLHT) problem for high-dimensional data under heteroscedasticity.
glhtbf_zzg2022(Y,G,n,p)
Y |
A list of |
G |
A known full-rank coefficient matrix ( |
n |
A vector of |
p |
The dimension of data. |
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
.
Zhang et al. (2022) proposed the following test statistic:
T_{ZZG}=\|\boldsymbol{C} \hat{\boldsymbol{\mu}}\|^2,
where \boldsymbol{C}=[(\boldsymbol{G D G}^\top)^{-1/2}\boldsymbol{G}]\otimes\boldsymbol{I}_p
with \boldsymbol{D}=\operatorname{diag}(1/n_1,\ldots,1/n_k)
, and \hat{\boldsymbol{\mu}}=(\bar{\boldsymbol{y}}_1^\top,\ldots,\bar{\boldsymbol{y}}_k^\top)^\top
with \bar{\boldsymbol{y}}_{i},i=1,\ldots,k
being the sample mean vectors.
They showed that under the null hypothesis, T_{ZZG}
and a chi-squared-type mixture have the same normal or non-normal limiting distribution.
A (list) object of S3
class htest
containing the following elements:
the p
-value of the test proposed by Zhang et al. (2022)
the test statistic proposed by Zhang et al. (2022).
the parameters used in Zhang et al. (2022)'s test.
estimated approximate degrees of freedom of Zhang et al. (2022)'s test.
Zhang J, Zhou B, Guo J (2022).
“Linear hypothesis testing in high-dimensional heteroscedastic one-way MANOVA: A normal reference L^2
-norm based test.”
Journal of Multivariate Analysis, 187, 104816.
doi:10.1016/j.jmva.2021.104816.
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_zzg2022(Y, G, n, p)