This function implements six two-sample simultaneous tests
on high-dimensional mean vectors and covariance matrices.
Let \(\mathbf{X} \in \mathbb{R}^p\) and \(\mathbf{Y} \in \mathbb{R}^p\)
be two \(p\)-dimensional populations with mean vectors
\((\boldsymbol{\mu}_1, \boldsymbol{\mu}_2)\) and covariance matrices
\((\mathbf{\Sigma}_1, \mathbf{\Sigma}_2)\), respectively.
The problem of interest is the simultaneous inference on the equality of
mean vectors and covariance matrices of the two populations:
$$H_0: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2 \ \text{ and }
\ \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2. $$
Suppose \(\{\mathbf{X}_1, \ldots, \mathbf{X}_{n_1}\}\) are i.i.d.
copies of \(\mathbf{X}\), and \(\{\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2}\}\)
are i.i.d. copies of \(\mathbf{Y}\). We denote
dataX=\((\mathbf{X}_1, \ldots, \mathbf{X}_{n_1})^\top\in\mathbb{R}^{n_1\times p}\)
and dataY=\((\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2})^\top\in\mathbb{R}^{n_2\times p}\).
simultest(dataX, dataY, method='pe.fisher', delta_mean=NULL, delta_cov=NULL)method the method type
stat the value of test statistic
pval the p-value for the test.
an \(n_1\) by \(p\) data matrix
an \(n_2\) by \(p\) data matrix
the method type (default = 'pe.fisher'); chosen from
'cauchy': the simultaneous test via Cauchy combination;
see simultest.cauchy for details.
'chisq': the simultaneous test via chi-squared approximation;
see simultest.chisq for details.
'fisher': the simultaneous test via Fisher's combination;
see simultest.fisher for details.
'pe.cauchy': the PE simultaneous test via Cauchy combination;
see simultest.pe.cauchy for details.
'pe.chisq': the PE simultaneous test via chi-squared approximation;
see simultest.pe.chisq for details.
'pe.fisher': the PE simultaneous test via Fisher's combination;
see simultest.pe.fisher for details.
the thresholding value used in the construction of
the PE component for the mean test statistic. It is needed only in PE methods such as
method='pe.cauchy', method='pe.chisq', and
method='pe.fisher'; see simultest.pe.cauchy,
simultest.pe.chisq,
and simultest.pe.fisher
for details. The default is NULL.
the thresholding value used in the construction of
the PE component for the covariance test statistic. It is needed only in PE methods such as
method='pe.cauchy', method='pe.chisq', and
method='pe.fisher'; see simultest.pe.cauchy,
simultest.pe.chisq,
and simultest.pe.fisher
for details. The default is NULL.
Chen, S. X. and Qin, Y. L. (2010). A two-sample test for high-dimensional data with applications to gene-set testing. Annals of Statistics, 38(2):808–835.
Li, J. and Chen, S. X. (2012). Two sample tests for high-dimensional covariance matrices. The Annals of Statistics, 40(2):908–940.
Yu, X., Li, D., and Xue, L. (2022). Fisher’s combined probability test for high-dimensional covariance matrices. Journal of the American Statistical Association, (in press):1–14.
Yu, X., Li, D., Xue, L., and Li, R. (2022). Power-enhanced simultaneous test of high-dimensional mean vectors and covariance matrices with application to gene-set testing. Journal of the American Statistical Association, (in press):1–14.
n1 = 100; n2 = 100; pp = 500
set.seed(1)
X = matrix(rnorm(n1*pp), nrow=n1, ncol=pp)
Y = matrix(rnorm(n2*pp), nrow=n2, ncol=pp)
simultest(X,Y)
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