# Normal-reference-test with two-cumulant (2-c) matched $\chi^2$-approximation for GLHT problem under heteroscedasticity proposed by Zhang et al. (2022) Zhang et al. (2022)'s test for general linear hypothesis testing (GLHT) problem for high-dimensional data under heteroscedasticity. ```r ZZG2022.GLHTBF.2cNRT(Y,G,n,p) ``` ## Arguments - `Y`: A list of $k$ data matrices. The $i$th element represents the data matrix ($n_i\times p$) from the $i$th population with each row representing a $p$-dimensional observation. - `G`: A known full-rank coefficient matrix ($q\times k$) with $\operatorname{rank}(\boldsymbol{G})< k$. - `n`: A vector of $k$ sample sizes. The $i$th element represents the sample size of group $i$, $n_i$. - `p`: The dimension of data. ## Returns 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 the following $k$ independent high-dimensional samples: $$ \boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma}_i,i=1,\ldots,k. $$ It is of interest to test the following GLHT problem: $$ H_0: \boldsymbol{G M}=\boldsymbol{0}, \quad \text { vs. } \; H_1: \boldsymbol{G M} \neq \boldsymbol{0}, $$ where $\boldsymbol{M}=(\boldsymbol{\mu}_1,\ldots,\boldsymbol{\mu}_k)^\top$ is a $k\times p$ matrix collecting $k$ mean vectors and $\boldsymbol{G}:q\times k$ is a known full-rank coefficient matrix with $\operatorname{rank}(\boldsymbol{G})

ZZG2022.GLHTBF.2cNRT function