Package 'Docovt'

Two-Sample Covariance Test by Cai, Liu and Xia (2013)

Description

Given two sets of data matrices X and Y, where X is an n1 rows and p cols matrix and Y is an n2 rows and p cols matrix, we conduct hypothesis testing of the covariance matrix between two samples. The null hypothesis is:

$H_0 : \Sigma_1 = \Sigma_2$

$\Sigma_1$ and $\Sigma_2$ are the sample covariance matrices of X and Y respectively. This test method is based on the test method proposed by Cai, Liu and Xia (2013). When the pval value is less than the significance coefficient (generally 0.05), the null hypothesis is rejected.

Usage

CLX(X,Y,alpha)

Arguments

X	A matrix of n1 by p
Y	A matrix of n2 by p
alpha	Significance level

Value

stat	a test statistic value.
pval	a test p_value.
power	a test power value.
FDR	a test FDR value.

References

Cai, T. T., Liu, W., and Xia, Y. (2013). Two-sample covariance matrix testing and support recovery in high-dimensional and sparse settings. Journal of the American Statistical Association, 108(501):265-277.

Guo, G., Wang, Q., Allison, J., & Qian, G. (2025). Accelerated distributed expectation-maximization algorithms for the parameter estimation in multivariate Gaussian mixture models. Applied Mathematical Modelling, 137, 115709.

Examples

## generate X and Y.
p= 500;  n1 = 100; n2 = 150
X=matrix(rnorm(n1*p), ncol=p)
Y=matrix(rnorm(n2*p), ncol=p)
## run test
CLX(X,Y,alpha=0.05)

---

One-Sample Covariance Test by Cai and Ma (2013)

Description

One-Sample Covariance Test by Cai and Ma (2013) Given data, it performs 1-sample test for Covariance where the null hypothesis is

$H_0 : \Sigma_n = \Sigma_0$

where $\Sigma_n$ is the covariance of data model and $\Sigma_0$ is a hypothesized covariance based on a procedure proposed by Cai and Ma (2013).

Usage

cm13(X, Sigma0, alpha)

Arguments

X	an $(n\times p)$ data matrix where each row is an observation.
Sigma0	a $(p\times p)$ given covariance matrix.
alpha	level of significance.

Details

Given data, it performs 1-sample test for Covariance where the null hypothesis is

$H_0 : \Sigma_n = \Sigma_0$

where $\Sigma_n$ is the covariance of data model and $\Sigma_0$ is a hypothesized covariance based on a procedure proposed by Cai and Ma (2013).

Value

a named list containing:

statistic

a test statistic value.

threshold

rejection criterion to be compared against test statistic.

reject

a logical; TRUE to reject null hypothesis, FALSE otherwise.

Examples

## Not run: 
## generate data from multivariate normal with trivial covariance.
p = 5;n=10
X=data= matrix(rnorm(n*p), ncol=p)
 alpha=0.05
Sigma0=diag(ncol(X))
cm13(X,Sigma0, alpha)

## End(Not run)

---

Two-Sample Covariance Test by Cai and Ma (2013)

Description

Given two sets of data, it performs 2-sample test for equality of covariance matrices where the null hypothesis is

$H_0 : \Sigma_1 = \Sigma_2$

where $\Sigma_1$ and $\Sigma_2$ represent true (unknown) covariance for each dataset based on a procedure proposed by Cai and Ma (2013). If statistic $>$ threshold, it rejects null hypothesis.

Usage

cmtwo(X, Y, alpha)

Arguments

X	an $(m\times p)$ matrix where each row is an observation from the first dataset.
Y	an $(n\times p)$ matrix where each row is an observation from the second dataset.
alpha	level of significance.

Value

a named list containing

statistic

a test statistic value.

threshold

rejection criterion to be compared against test statistic.

reject

a logical; TRUE to reject null hypothesis, FALSE otherwise.

References

Guo, G., Qian, G., & Zhu, L. (2022). A scalable quasi-newton estimation algorithm for dynamic generalized linear models. Journal of Nonparametric Statistics, 34, 917-939. doi:10.1080/10485252.2022.2085263.

Guo, G., & Zhao, W. (2021). Schwarz method for financial engineering. Journal of Computational Mathematics, 39(4), 538-555.

You, W., Yang, Z., Guo, G., Wan, X.-F., & Ji, G. (2019). Prediction of DNA-binding proteins by interaction fusion feature representation and selective ensemble. Knowledge-Based Systems, 163, 598-610.

Examples

## generate 2 datasets from multivariate normal with identical covariance.
p= 5;  n1 = 100; n2 = 150; alpha=0.05
X=data1 = matrix(rnorm(n1*p), ncol=p)
Y=data2 = matrix(rnorm(n2*p), ncol=p)

# run test
cmtwo(X, Y, alpha)

---

corneal

Description

This dataset was acquired during a keratoconus study, a collaborative project involving Ms.Nancy Tripoli and Dr.Kenneth L.Cohen of Department of Ophthalmology at the University of North Carolina, Chapel Hill. The fitted feature vectors for the complete corneal surface dataset collectively into a feature matrix with dimensions of 150 x 2000.

Usage

data(corneal)

Format

## 'corneal' A data frame with 150 observations on the following 4 groups.

normal group1

row 1 to row 43 in total 43 rows of the feature matrix correspond to observations from the normal group

unilateral suspect group2

row 44 to row 57 in total 14 rows of the feature matrix correspond to observations from the unilateral suspect group

suspect map group3

row 58 to row 78 in total 21 of the feature matrix correspond to observations from the suspect map group

clinical keratoconus group4

row 79 to row 150 in total 72 of the feature matrix correspond to observations from the clinical keratoconus group

Examples

data(corneal)
dim(corneal)
group1 <- as.matrix(corneal[1:43, ]) ## normal group
dim(group1)
group2 <- as.matrix(corneal[44:57, ]) ## unilateral suspect group
dim(group2)
group3 <- as.matrix(corneal[58:78, ]) ## suspect map group
dim(group3)
group4 <- as.matrix(corneal[79:150, ]) ## clinical keratoconus group
dim(group4)

---

COVID19

Description

A COVID19 data set from NCBI with ID GSE152641. The data set profiled peripheral blood from 24 healthy controls and 62 prospectively enrolled patients with community-acquired lower respiratory tract infection by SARS-COV-2 within the first 24 hours of hospital admission using RNA sequencing.

Usage

data(COVID19)

Format

## 'COVID19' A data frame with 86 observations on the following 2 groups.

healthy group1

row 2 to row 19, and row 82 to 87, in total 24 healthy controls

patients group2

row 20 to 81, in total 62 prospectively enrolled patients

Examples

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

---

Distributed two-sample covariance matrix testing

Description

Distributed two-sample covariance matrix testing Given two sets of data matrices X and Y, where X is an n1 rows and p cols matrix and Y is an n2 rows and p cols matrix, we conduct hypothesis testing of the covariance matrix between two samples. The null hypothesis is:

$H_0 : \Sigma_1 = \Sigma_2$

$\Sigma_1$ and $\Sigma_2$ are the sample covariance matrices of X and Y respectively. This test method is based on the test method proposed by Cai, Liu and Xia (2013). When the pval value is less than the significance coefficient (generally 0.05), the null hypothesis is rejected.

Usage

DCLX(X, Y, K)

Arguments

X	A matrix of n1 by p.
Y	A matrix of n2 by p.
K	Number of blocks.

Details

Given two sets of data matrices X and Y, where X is an n1 rows and p cols matrix and Y is an n2 rows and p cols matrix, we conduct hypothesis testing of the covariance matrix between two samples. The null hypothesis is:

$H_0 : \Sigma_1 = \Sigma_2$

$\Sigma_1$ and $\Sigma_2$ are the sample covariance matrices of X and Y respectively. This test method is based on the test method proposed by Cai, Liu and Xia (2013). When the pval value is less than the significance coefficient (generally 0.05), the null hypothesis is rejected.

Value

stat	a test statistic value.
pval	a test p_value.

Examples

## Not run: 
## generate data from multivariate normal with trivial covariance.
##generate X and Y.
p= 500;  n1 = 100; n2 = 150;K=10
X=matrix(rnorm(n1*p), ncol=p)
Y=matrix(rnorm(n2*p), ncol=p)
## run test
DCLX(X,Y,K)

## End(Not run)

---

Distributed two-sample covariance matrix testing

Description

Distributed two-sample covariance matrix testing Given two sets of data matrices X and Y, where X is an n1 rows and p cols matrix and Y is an n2 rows and p cols matrix, we conduct hypothesis testing of the covariance matrix between two samples. The null hypothesis is:

$H_0 : \Sigma_1 = \Sigma_2$

$\Sigma_1$ and $\Sigma_2$ are the sample covariance matrices of X and Y respectively. This test method is based on the test method proposed by Li and Chen (2012). When the pval value is less than the significance coefficient (generally 0.05), the null hypothesis is rejected.

Usage

DLC(X, Y, K)

Arguments

X	A matrix of n1 by p.
Y	A matrix of n2 by p.
K	Number of blocks.

Details

Given two sets of data matrices X and Y, where X is an n1 rows and p cols matrix and Y is an n2 rows and p cols matrix, we conduct hypothesis testing of the covariance matrix between two samples. The null hypothesis is:

$H_0 : \Sigma_1 = \Sigma_2$

$\Sigma_1$ and $\Sigma_2$ are the sample covariance matrices of X and Y respectively. This test method is based on the test method proposed by Li and Chen (2012). When the pval value is less than the significance coefficient (generally 0.05), the null hypothesis is rejected.

Value

stat	a test statistic value.
pval	a test p_value.

Examples

## Not run: 
## generate data from multivariate normal with trivial covariance.
##generate X and Y.
p= 500;  n1 = 100; n2 = 150;K=10
X=matrix(rnorm(n1*p), ncol=p)
Y=matrix(rnorm(n2*p), ncol=p)
## run test
DLC(X,Y,K)

## End(Not run)

---

Distributed two-sample covariance matrix testing

Description

Distributed two-sample covariance matrix testing Given two sets of data matrices X and Y, where X is an n1 rows and p cols matrix and Y is an n2 rows and p cols matrix, we conduct hypothesis testing of the covariance matrix between two samples. The null hypothesis is:

$H_0 : \Sigma_1 = \Sigma_2$

$\Sigma_1$ and $\Sigma_2$ are the sample covariance matrices of X and Y respectively. This test method is based on the test method proposed by Yu, Li and Xue (2022). When the pval value is less than the significance coefficient (generally 0.05), the null hypothesis is rejected.

Usage

DPEC(X, Y, K)

Arguments

X	A matrix of n1 by p.
Y	A matrix of n2 by p.
K	Number of blocks.

Details

Given two sets of data matrices X and Y, where X is an n1 rows and p cols matrix and Y is an n2 rows and p cols matrix, we conduct hypothesis testing of the covariance matrix between two samples. The null hypothesis is:

$H_0 : \Sigma_1 = \Sigma_2$

$\Sigma_1$ and $\Sigma_2$ are the sample covariance matrices of X and Y respectively. This test method is based on the test method proposed by Yu, Li and Xue (2022). When the pval value is less than the significance coefficient (generally 0.05), the null hypothesis is rejected.

Value

stat	a test statistic value.
pval	a test p_value.

Examples

## Not run: 
## generate data from multivariate normal with trivial covariance.
##generate X and Y.
p= 500;  n1 = 100; n2 = 150;K=10
X=matrix(rnorm(n1*p), ncol=p)
Y=matrix(rnorm(n2*p), ncol=p)
## run test
DPEC(X,Y,K)

## End(Not run)

---

Distributed two-sample covariance matrix testing

Description

Distributed two-sample covariance matrix testing Given two sets of data matrices X and Y, where X is an n1 rows and p cols matrix and Y is an n2 rows and p cols matrix, we conduct hypothesis testing of the covariance matrix between two samples. The null hypothesis is:

$H_0 : \Sigma_1 = \Sigma_2$

$\Sigma_1$ and $\Sigma_2$ are the sample covariance matrices of X and Y respectively. This test method is based on the test method proposed by Yu, Li, Xue and Li (2022). When the pval value is less than the significance coefficient (generally 0.05), the null hypothesis is rejected.

Usage

DPECO(X, Y, delta = NULL, K)

Arguments

X	A matrix of n1 by p.
Y	A matrix of n2 by p.
delta	A scalar used as the threshold for building PE components, usually the default value.
K	Number of blocks.

Details

Given two sets of data matrices X and Y, where X is an n1 rows and p cols matrix and Y is an n2 rows and p cols matrix, we conduct hypothesis testing of the covariance matrix between two samples. The null hypothesis is:

$H_0 : \Sigma_1 = \Sigma_2$

$\Sigma_1$ and $\Sigma_2$ are the sample covariance matrices of X and Y respectively. This test method is based on the test method proposed by Yu, Li, Xue and Li (2022). When the pval value is less than the significance coefficient (generally 0.05), the null hypothesis is rejected.

Value

stat	a test statistic value.
pval	a test p_value.

Examples

## Not run: 
## generate data from multivariate normal with trivial covariance.
##generate X and Y.
p= 500;  n1 = 100; n2 = 150;K=10
X=matrix(rnorm(n1*p), ncol=p)
Y=matrix(rnorm(n2*p), ncol=p)
## run test
DPECO(X,Y,K)

## End(Not run)

---

Distributed two-sample covariance matrix testing

Description

Distributed two-sample covariance matrix testing Given two sets of data matrices X and Y, where X is an n1 rows and p cols matrix and Y is an n2 rows and p cols matrix, we conduct hypothesis testing of the covariance matrix between two samples. The null hypothesis is:

$H_0 : \Sigma_1 = \Sigma_2$

$\Sigma_1$ and $\Sigma_2$ are the sample covariance matrices of X and Y respectively. This test method is based on the test method proposed by Yu, Li and Xue (2022). When the pval value is less than the significance coefficient (generally 0.05), the null hypothesis is rejected.

Usage

DPEF(X, Y, K)

Arguments

X	A matrix of n1 by p.
Y	A matrix of n2 by p.
K	Number of blocks.

Details

Given two sets of data matrices X and Y, where X is an n1 rows and p cols matrix and Y is an n2 rows and p cols matrix, we conduct hypothesis testing of the covariance matrix between two samples. The null hypothesis is:

$H_0 : \Sigma_1 = \Sigma_2$

$\Sigma_1$ and $\Sigma_2$ are the sample covariance matrices of X and Y respectively. This test method is based on the test method proposed by Yu, Li and Xue (2022). When the pval value is less than the significance coefficient (generally 0.05), the null hypothesis is rejected.

Value

stat	a test statistic value.
pval	a test p_value.

Examples

## Not run: 
## generate data from multivariate normal with trivial covariance.
##generate X and Y.
p= 500;  n1 = 100; n2 = 150
X=matrix(rnorm(n1*p), ncol=p)
Y=matrix(rnorm(n2*p), ncol=p)
## run test
DPEF(X,Y)

## End(Not run)

---

Distributed two-sample covariance matrix testing

Description

Distributed two-sample covariance matrix testing Given two sets of data matrices X and Y, where X is an n1 rows and p cols matrix and Y is an n2 rows and p cols matrix, we conduct hypothesis testing of the covariance matrix between two samples. The null hypothesis is:

$H_0 : \Sigma_1 = \Sigma_2$

$\Sigma_1$ and $\Sigma_2$ are the sample covariance matrices of X and Y respectively. This test method is based on the test method proposed by Cai, Liu and Xia (2013). When the pval value is less than the significance coefficient (generally 0.05), the null hypothesis is rejected.

Usage

DSNET(X, Y, K, B = 200, mc.cores = 2L)

Arguments

X	A matrix of n1 by p.
Y	A matrix of n2 by p.
K	Number of blocks.
B	number of bootstrap.
mc.cores	Number of cores for parallel computing. Default is 2.

Details

Given two sets of data matrices X and Y, where X is an n1 rows and p cols matrix and Y is an n2 rows and p cols matrix, we conduct hypothesis testing of the covariance matrix between two samples. The null hypothesis is:

$H_0 : \Sigma_1 = \Sigma_2$

$\Sigma_1$ and $\Sigma_2$ are the sample covariance matrices of X and Y respectively. This test method is based on the test method proposed by Cai, Liu and Xia (2013). When the pval value is less than the significance coefficient (generally 0.05), the null hypothesis is rejected.

Value

stat	a test statistic value.
pval	a test p_value.

Examples

## Not run: 
## generate data from multivariate normal with trivial covariance.
##generate X and Y.
p= 500;  n1 = 100; n2 = 150;B=300;K=10
X=matrix(rnorm(n1*p), ncol=p)
Y=matrix(rnorm(n2*p), ncol=p)
## run test
DSNET(X,Y,K,B)

## End(Not run)

---

Two-Sample Covariance Test by Li and Chen (2012)

Description

Given two sets of data matrices X and Y, where X is an n1 rows and p cols matrix and Y is an n2 rows and p cols matrix, we conduct hypothesis testing of the covariance matrix between two samples. The null hypothesis is:

$H_0 : \Sigma_1 = \Sigma_2$

$\Sigma_1$ and $\Sigma_2$ are the sample covariance matrices of X and Y respectively. This test method is based on the test method proposed by Li and Chen (2012). When the pval value is less than the significance coefficient (generally 0.05), the null hypothesis is rejected.

Usage

LC(X,Y,delta_sigma = NULL,alpha)

Arguments

X	A matrix of n1 by p
Y	A matrix of n2 by p
delta_sigma	A positive definite matrix
alpha	Significance level

Value

stat	a test statistic value.
pval	a test p_value.
power	a test power value.
FDR	a test FDR value.

References

Li, J. and Chen, S. X. (2012). Two sample tests for high-dimensional covariance matrices. The Annals of Statistics, 40(2):908-940.

Zhang, C., & Guo, G. (2025). The optimal subset estimation of distributed redundant data. IAENG International Journal of Applied Mathematics, 55(2), 270–277.

Examples

## generate X and Y.
p= 500;  n1 = 100; n2 = 150
X=matrix(rnorm(n1*p), ncol=p)
Y=matrix(rnorm(n2*p), ncol=p)
## run test
LC(X,Y,alpha=0.05)

---

miRNA

Description

A three factor level variable corresponding to cancer type

Usage

data(miRNA)

Format

Dataframe with 21 samples and 537 variables

columns

variables

rows

samples

Examples

data(miRNA)

---

Two-Sample Covariance Test by Yu, Li and Xue (2022)

Description

Given two sets of data matrices X and Y, where X is an n1 rows and p cols matrix and Y is an n2 rows and p cols matrix,, we conduct hypothesis testing of the covariance matrix between two samples. The null hypothesis is:

$H_0 : \Sigma_1 = \Sigma_2$

$\Sigma_1$ and $\Sigma_2$ are the sample covariance matrices of X and Y respectively. This test method is based on the test method proposed by Yu, Li and Xue (2022). When the pval value is less than the significance coefficient (generally 0.05), the null hypothesis is rejected.

Usage

PEC(X,Y,alpha)

Arguments

X	A matrix of n1 by p
Y	A matrix of n2 by p
alpha	Significance level

Value

stat	a test statistic value.
pval	a test p_value.
power	a test power value.
FDR	a test FDR value.

References

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.

Liu, S., & Guo, G. (2025). Sparse online principal component for Laplace factor models. IAENG International Journal of Applied Mathematics, 55(6), 1588–1592.

Examples

## generate X and Y.
p= 500;  n1 = 100; n2 = 150
X=matrix(rnorm(n1*p), ncol=p)
Y=matrix(rnorm(n2*p), ncol=p)
## run test
PEC(X,Y,alpha=0.05)

---

Two-Sample Covariance Test by Yu, Li, Xue and Li(2022)

Description

Given two sets of data matrices X and Y, where X is an n1 rows and p cols matrix and Y is an n2 rows and p cols matrix, we conduct hypothesis testing of the covariance matrix between two samples. The null hypothesis is:

$H_0 : \Sigma_1 = \Sigma_2$

$\Sigma_1$ and $\Sigma_2$ are the sample covariance matrices of X and Y respectively. This test method is based on the test method proposed by Yu, Li, Xue and Li (2022). When the pval value is less than the significance coefficient (generally 0.05), the null hypothesis is rejected.

Usage

PECO(X,Y,delta = NULL,alpha)

Arguments

X	A matrix of n1 by p
Y	A matrix of n2 by p
alpha	Significance level
delta	A scalar used as the threshold for building PE components, usually the default value

Value

stat	a test statistic value.
pval	a test p_value.
power	a test power value.
FDR	a test FDR value.

References

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.

Jin, Y., Chang, D., & Guo, G. (2025). Sparse online principal component for skew factor model. IAENG International Journal of Computer Science, 52(9), 3429–3434.

Examples

## generate X and Y.
p= 500;  n1 = 100; n2 = 150
X=matrix(rnorm(n1*p), ncol=p)
Y=matrix(rnorm(n2*p), ncol=p)
## run test
PECO(X,Y,alpha=0.05)

---

Two-Sample Covariance Test by Yu, Li and Xue (2022)

Description

Given two sets of data matrices X and Y, where X is an n1 rows and p cols matrix and Y is an n2 rows and p cols matrix,, we conduct hypothesis testing of the covariance matrix between two samples. The null hypothesis is:

$H_0 : \Sigma_1 = \Sigma_2$

$\Sigma_1$ and $\Sigma_2$ are the sample covariance matrices of X and Y respectively. This test method is based on the test method proposed by Yu, Li and Xue (2022). When the pval value is less than the significance coefficient (generally 0.05), the null hypothesis is rejected.

Usage

PEF(X,Y,alpha)

Arguments

X	A matrix of n1 by p
Y	A matrix of n2 by p
alpha	Significance level

Value

stat	a test statistic value.
pval	a test p_value.
power	a test power value.
FDR	a test FDR value.

References

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.

Guo, G., Song, H., & Zhu, L. (2024). The COR criterion for optimal subset selection in distributed estimation. Statistics and Computing, 34, 163-176.

Examples

## generate X and Y.
p= 500;  n1 = 100; n2 = 150
X=matrix(rnorm(n1*p), ncol=p)
Y=matrix(rnorm(n2*p), ncol=p)
## run test
PEF(X,Y,alpha=0.05)

---

Two-Sample Covariance Test

Description

Two-Sample Covariance Test Given two sets of data matrices X and Y, where X is an n1 rows and p cols matrix and Y is an n2 rows and p cols matrix, we conduct hypothesis testing of the covariance matrix between two samples. The null hypothesis is:

$H_0 : \Sigma_1 = \Sigma_2$

$\Sigma_1$ and $\Sigma_2$ are the sample covariance matrices of X and Y respectively. This test method is based on the test method proposed by Cai, Liu and Xia (2013). When the pval value is less than the significance coefficient (generally 0.05), the null hypothesis is rejected.

Usage

SNET(X, Y, B = 200, mc.cores = 2L, alpha = 0.05)

Arguments

X	A matrix of n1 by p.
Y	A matrix of n2 by p.
B	number of bootstrap.
mc.cores	Number of cores for parallel computing. Default is 2.
alpha	Significance level.

Details

Given two sets of data matrices X and Y, where X is an n1 rows and p cols matrix and Y is an n2 rows and p cols matrix, we conduct hypothesis testing of the covariance matrix between two samples. The null hypothesis is:

$H_0 : \Sigma_1 = \Sigma_2$

$\Sigma_1$ and $\Sigma_2$ are the sample covariance matrices of X and Y respectively. This test method is based on the test method proposed by Cai, Liu and Xia (2013). When the pval value is less than the significance coefficient (generally 0.05), the null hypothesis is rejected.

Value

stat	a test statistic value.
pval	a test p_value.

Examples

## Not run: 
## generate data from multivariate normal with trivial covariance.
##generate X and Y.
p= 500;  n1 = 100; n2 = 150;B=300
X=matrix(rnorm(n1*p), ncol=p)
Y=matrix(rnorm(n2*p), ncol=p)
## run test
SNET(X,Y,B,alpha = 0.05)

## End(Not run)

---

One Sample Covariance Test by Srivastava, Yanagihara, and Kubokawa (2014)

Description

Given data, it performs 1-sample test for Covariance where the null hypothesis is

$H_0 : \Sigma_n = \Sigma_0$

where $\Sigma_n$ is the covariance of data model and $\Sigma_0$ is a hypothesized covariance based on a procedure proposed by Srivastava, Yanagihara, and Kubokawa (2014).

Usage

syk(data, Sigma0, alpha)

Arguments

data	an $(n\times p)$ data matrix where each row is an observation.
Sigma0	a $(p\times p)$ given covariance matrix.
alpha	level of significance.

Value

a named list containing

statistic

a test statistic value.

threshold

rejection criterion to be compared against test statistic.

reject

a logical; TRUE to reject null hypothesis, FALSE otherwise.

Examples

## Not run: 
## generate data from multivariate normal with trivial covariance.
p = 5;n=10
data = matrix(rnorm(n*p), ncol=p)
 alpha=0.05
Sigma0=diag(ncol(data))
 run the test
syk(data, Sigma0, alpha)

## End(Not run)

---