Bai and Saranadasa (1996)'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 Bai and Saranadasa (1996)
- p.value
the \(p\)-value of the test proposed by Bai and Saranadasa (1996).
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 $$H_{0}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2\; \operatorname{versus}\; H_{1}: \boldsymbol{\mu}_1 \neq \boldsymbol{\mu}_2.$$ Bai and Saranadasa (1996) proposed the following centralised \(L^2\)-norm-based test statistic: $$T_{BS} = \frac{n_1n_2}{n} \|\bar{\boldsymbol{y}}_1 - \bar{\boldsymbol{y}}_2\|^2-\operatorname{tr}(\hat{\boldsymbol{\Sigma}}),$$ where \(\bar{\boldsymbol{y}}_{i},i=1,2\) are the sample mean vectors and \(\hat{\boldsymbol{\Sigma}}\) is the pooled sample covariance matrix. They showed that under the null hypothesis, \(T_{BS}\) is asymptotically normally distributed.
References
Bai Z, Saranadasa H (1996). “Effect of high dimension: by an example of a two sample problem.” Statistica Sinica, 311--329. https://www.jstor.org/stable/24306018.
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_bs1996(y1, y2)
#>
#>
#>
#> data:
#> statistic = 0.038109, p-value = 0.4848
#>