glht_zgz2017 {HDNRA} | R Documentation |
Zhang et al. (2017)'s test for general linear hypothesis testing (GLHT) problem for high-dimensional data with assuming that underlying covariance matrices are the same.
glht_zgz2017(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=1,\ldots,k.
It is of interest to test the following GLHT problem:
H_0: \boldsymbol{G M}=\boldsymbol{0}, \quad \text { vs. } \quad 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. (2017) proposed the following test statistic:
T_{ZGZ}=\|\boldsymbol{C \hat{\mu}}\|^2,
where \boldsymbol{C}=[(\boldsymbol{G D G}^\top)^{-1/2}\boldsymbol{G}]\otimes\boldsymbol{I}_p
, 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 and \boldsymbol{D}=\operatorname{diag}(1/n_1,\ldots,1/n_k)
.
They showed that under the null hypothesis, T_{ZGZ}
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 test statistic proposed by Zhang et al. (2017)
the p
-value of the test proposed by Zhang et al. (2017).
the parameters used in Zhang et al. (2017)'s test.
estimated approximate degrees of freedom of Zhang et al.(2017)'s test.
Zhang J, Guo J, Zhou B (2017). “Linear hypothesis testing in high-dimensional one-way MANOVA.” Journal of Multivariate Analysis, 155, 200–216. doi:10.1016/j.jmva.2017.01.002.
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)
y <- (-2 * sqrt(1 - rho) + sqrt(4 * (1 - rho) + 4 * p * rho)) / (2 * p)
x <- y + sqrt((1 - rho))
Gamma <- matrix(rep(y, p * p), nrow = p)
diag(Gamma) <- rep(x, p)
Y <- list()
for (g in 1:k) {
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))
glht_zgz2017(Y, G, n, p)