glht_zzz2022 {HDNRA} | R Documentation |
Test proposed by Zhu et al. (2022)
Description
Zhu et al. (2022)'s test for general linear hypothesis testing (GLHT) problem for high-dimensional data with assuming that underlying covariance matrices are the same.
Usage
glht_zzz2022(Y,X,C)
Arguments
Y |
An |
X |
A known |
C |
A known matrix of size |
Details
A high-dimensional linear regression model can be expressed as
where is a
unknown parameter matrix and
is an
error matrix.
It is of interest to test the following GLHT problem
Zhu et al. (2022) proposed the following test statistic:
where and
are the variation matrices due to the hypothesis and error, respectively, and
is the diagonal matrix with the diagonal elements of
.
They showed that under the null hypothesis,
and a chi-squared-type mixture have the same limiting distribution.
Value
A (list) object of S3
class htest
containing the following elements:
- p.value
the
-value of the test proposed by Zhu et al. (2022)
- statistic
the test statistic proposed by Zhu et al. (2022).
- df
estimated approximate degrees of freedom of Zhu et al. (2022)'s test.
References
Zhu T, Zhang L, Zhang J (2023). “Hypothesis Testing in High-Dimensional Linear Regression: A Normal Reference Scale-Invariant Test.” Statistica Sinica. doi:10.5705/ss.202020.0362.
Examples
set.seed(1234)
k <- 3
q <- k - 1
p <- 50
n <- c(25, 30, 40)
rho <- 0.01
Theta <- matrix(rep(0, k * p), nrow = k)
X <- matrix(c(rep(1, n[1]), rep(0, sum(n)), rep(1, n[2]), rep(0, sum(n)), rep(1, n[3])),
ncol = k, nrow = sum(n)
)
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)
U <- matrix(ncol = sum(n), nrow = p)
for (i in 1:sum(n)) {
U[, i] <- rnorm(p, 0, 1)
}
Y <- X %*% Theta + t(U) %*% Gamma
C <- cbind(diag(q), -rep(1, q))
glht_zzz2022(Y, X, C)