SKK2013.TSBF.NABT: Normal-approximation-based test for two-sample BF problem...

In HDNRA: High-Dimensional Location Testing with Normal-Reference Approaches

Normal-approximation-based test for two-sample BF problem proposed by Srivastava et al. (2013)

Description

Srivastava et al. (2013)'s test for testing equality of two-sample high-dimensional mean vectors without assuming that two covariance matrices are the same.

Usage

SKK2013.TSBF.NABT(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.

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.

Srivastava et al. (2013) proposed the following test statistic:

T_{SKK} = \frac{(\bar{\boldsymbol{y}}_1 - \bar{\boldsymbol{y}}_2)^\top \hat{\boldsymbol{D}}^{-1}(\bar{\boldsymbol{y}}_1 - \bar{\boldsymbol{y}}_2) - p}{\sqrt{2 \widehat{\operatorname{Var}}(\hat{q}_n) c_{p,n}}},

where \bar{\boldsymbol{y}}_{i},i=1,2 are the sample mean vectors, \hat{\boldsymbol{D}}=\hat{\boldsymbol{D}}_1/n_1+\hat{\boldsymbol{D}}_2/n_2 with \hat{\boldsymbol{D}}_i,i=1,2 being the diagonal matrices consisting of only the diagonal elements of the sample covariance matrices. \widehat{\operatorname{Var}}(\hat{q}_n) is given by equation (1.18) in Srivastava et al. (2013), and c_{p, n} is the adjustment coefficient proposed by Srivastava et al. (2013). They showed that under the null hypothesis, T_{SKK} is asymptotically normally distributed.

Value

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

References

\insertRef

Srivastava_2013HDNRA

Examples

library("HDNRA")
data("COVID19")
dim(COVID19)
group1 <- as.matrix(COVID19[c(2:19, 82:87), ]) ## healthy group
group2 <- as.matrix(COVID19[-c(1:19, 82:87), ]) ## COVID-19 patients
SKK2013.TSBF.NABT(group1,group2)

HDNRA documentation built on Oct. 30, 2024, 9:28 a.m.