Normal-reference-test with three-cumulant (3-c) matched χ^2-approximation for two-sample BF problem proposed by Zhang and Zhu (2022)

Zhang and Zhu (2022)'s test for testing equality of two-sample high-dimensional mean vectors without assuming that two covariance matrices are the same.

Usage

ZZ2022.TSBF.3cNRT(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 $$H_{0}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2\; \operatorname{versus}\; H_{1}: \boldsymbol{\mu}_1 \neq \boldsymbol{\mu}_2.$$ Zhang and Zhu (2022) proposed the following test statistic: $$T_{ZZ} = \|\bar{\boldsymbol{y}}_1 - \bar{\boldsymbol{y}}_2\|^2-\operatorname{tr}(\hat{\boldsymbol{\Omega}}_n),$$ where \(\bar{\boldsymbol{y}}_{i},i=1,2\) are the sample mean vectors and \(\hat{\boldsymbol{\Omega}}_n\) is the estimator of \(\operatorname{Cov}(\bar{\boldsymbol{y}}_1-\bar{\boldsymbol{y}}_2)\). They showed that under the null hypothesis, \(T_{ZZ}\) and a chi-squared-type mixture have the same normal or non-normal limiting distribution.

References

Zhang J, Zhu T (2022). “A further study on Chen-Qin’s test for two-sample Behrens--Fisher problems for high-dimensional data.” Journal of Statistical Theory and Practice, 16(1), 1. doi:10.1007/s42519-021-00232-w .

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
ZZ2022.TSBF.3cNRT(group1, group2)
#> 
#> Results of Hypothesis Test
#> --------------------------
#> 
#> Test name:                       Zhang and Zhu (2022)'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_ZZ = 10073497402
#> 
#> Approximation method to the      3-c matched chi^2-approximation
#> null distribution of T_ZZ: 
#> 
#> Approximation parameter(s):      df    =  1.912900e+00
#>                                  beta0 = -2.649726e+09
#>                                  beta1 =  1.385153e+09
#> 
#> P-value:                         0.009152364
#>