simultest {PEtests}

Two-sample simultaneous tests on high-dimensional mean vectors and covariance matrices

Description

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}.

Usage

simultest(dataX, dataY, method='pe.fisher', delta_mean=NULL, delta_cov=NULL)

Arguments

dataX

an n_1 by p data matrix

dataY

an n_2 by p data matrix

method

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.

delta_mean

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.

delta_cov

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.

Value

method the method type

stat the value of test statistic

pval the p-value for the test.

References

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.

Examples

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)