Srivastava and Du (2008)'s test for testing equality of two-sample high-dimensional mean vectors with assuming that two covariance matrices are the same.
Arguments
- y1
The data matrix (p by n1) from the first population. Each column represents a \(p\)-dimensional observation.
- y2
The data matrix (p by n2) from the first population. Each column represents a \(p\)-dimensional observation.
Value
A (list) object of S3 class htest containing the following elements:
- statistic
the test statistic proposed by Srivastava and Du (2008).
- p.value
the \(p\)-value of the test proposed by Srivastava and Du (2008).
- cpn
the adjustment coefficient proposed by Srivastava and Du (2008).
Details
Suppose we have two independent high-dimensional samples: $$ \boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma},i=1,2. $$ The primary object is to test $$H_{0}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2\; \operatorname{versus}\; H_{1}: \boldsymbol{\mu}_1 \neq \boldsymbol{\mu}_2.$$ Srivastava and Du (2008) proposed the following test statistic: $$T_{SD} = \frac{n^{-1}n_1n_2(\bar{\boldsymbol{y}}_1 - \bar{\boldsymbol{y}}_2)^\top \boldsymbol{D}_S^{-1}(\bar{\boldsymbol{y}}_1 - \bar{\boldsymbol{y}}_2) - \frac{(n-2)p}{n-4}}{\sqrt{2 \left[\operatorname{tr}(\boldsymbol{R}^2) - \frac{p^2}{n-2}\right] c_{p, n}}},$$ where \(\bar{\boldsymbol{y}}_{i},i=1,2\) are the sample mean vectors, \(\boldsymbol{D}_S\) is the diagonal matrix of sample variance, \(\boldsymbol{R}\) is the sample correlation matrix and \(c_{p, n}\) is the adjustment coefficient proposed by Srivastava and Du (2008). They showed that under the null hypothesis, \(T_{SD}\) is asymptotically normally distributed.
References
Srivastava MS, Du M (2008). “A test for the mean vector with fewer observations than the dimension.” Journal of Multivariate Analysis, 99(3), 386--402. doi:10.1016/j.jmva.2006.11.002 .
Examples
set.seed(1234)
n1 <- 20
n2 <- 30
p <- 50
mu1 <- t(t(rep(0, p)))
mu2 <- mu1
rho <- 0.1
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)
Z1 <- matrix(rnorm(n1 * p, mean = 0, sd = 1), p, n1)
Z2 <- matrix(rnorm(n2 * p, mean = 0, sd = 1), p, n2)
y1 <- Gamma %*% Z1 + mu1 %*% (rep(1, n1))
y2 <- Gamma %*% Z2 + mu2 %*% (rep(1, n2))
ts_sd2008(y1, y2)
#>
#>
#>
#> data:
#> statistic = -0.41868, cpn = 1.3466, p-value = 0.6623
#>