Normal-reference-test with two-cumulant (2-c) matched $\chi^2$-approximation for two-sample BF problem proposed by Zhu et al. (2023)

Zhu et al. (2023)'s test for testing equality of two-sample high-dimensional mean vectors without assuming that two covariance matrices are the same.

Usage

ZWZ2023.TSBF.2cNRT(y1, y2)

Arguments

y1

The data matrix (\(n_1 \times p\)) from the first population. Each row represents a \(p\)-dimensional observation.

y2

The data matrix (\(n_2 \times p\)) from the second population. Each row represents a \(p\)-dimensional observation.

Value

A list of class "NRtest" containing the results of the hypothesis test. See the help file for NRtest.object for details.

Details

Suppose we have two independent high-dimensional samples: $$ \boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma}_i,\; i=1,2. $$ The primary object is to test $$H_{0}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2\; \operatorname{versus}\; H_{1}: \boldsymbol{\mu}_1 \neq \boldsymbol{\mu}_2.$$ Zhu et al. (2023) proposed the following test statistic: $$T_{ZWZ}=\frac{n_1n_2n^{-1}\|\bar{\boldsymbol{y}}_1-\bar{\boldsymbol{y}}_2\|^2}{\operatorname{tr}(\hat{\boldsymbol{\Omega}}_n)},$$ where \(\bar{\boldsymbol{y}}_{i},i=1,2\) are the sample mean vectors and \(\hat{\boldsymbol{\Omega}}_n\) is the estimator of \(\operatorname{Cov}[(n_1n_2/n)^{1/2}(\bar{\boldsymbol{y}}_1-\bar{\boldsymbol{y}}_2)]\). They showed that under the null hypothesis, \(T_{ZWZ}\) and an F-type mixture have the same normal or non-normal limiting distribution.

References

zhu2022twoHDNRA

Examples

library("HDNRA")
data("COVID19")
dim(COVID19)
#> [1]    87 20460
group1 <- as.matrix(COVID19[c(2:19, 82:87), ]) ## healthy group
group2 <- as.matrix(COVID19[-c(1:19, 82:87), ]) ## COVID-19 patients
ZWZ2023.TSBF.2cNRT(group1, group2)
#> 
#> Results of Hypothesis Test
#> --------------------------
#> 
#> Test name:                       Zhu et al. (2023)'s test
#> 
#> Null Hypothesis:                 Difference between two mean vectors is 0
#> 
#> Alternative Hypothesis:          Difference between two mean vectors is not 0
#> 
#> Data:                            group1 and group2
#> 
#> Sample Sizes:                    n1 = 24
#>                                  n2 = 62
#> 
#> Sample Dimension:                20460
#> 
#> Test Statistic:                  T[ZWZ] = 4.1877
#> 
#> Approximation method to the      2-c matched chi^2-approximation
#> null distribution of T[ZWZ]: 
#> 
#> Approximation parameter(s):      df1 =   2.7324
#>                                  df2 = 171.7596
#> 
#> P-value:                         0.008672887
#>