epval_Cai2014 {highmean}

Empirical Permutation- or Resampling-Based p-value of the Test Proposed by Cai et al (2014)

Description

Calculates p-value of the test for testing equality of two-sample high-dimensional mean vectors proposed by Cai et al (2014) based on permutation or parametric bootstrap resampling.

Usage

epval_Cai2014(sam1, sam2, eq.cov = TRUE, n.iter = 1000, cov1.est, cov2.est,
              bandwidth1, bandwidth2, cv.fold = 5, norm = "F", seeds)

Arguments

sam1

an n1 by p matrix from sample population 1. Each row represents a p-dimensional sample.

sam2

an n2 by p matrix from sample population 2. Each row represents a p-dimensional sample.

eq.cov

a logical value. The default is TRUE, indicating that the two sample populations have same covariance; otherwise, the covariances are assumed to be different. If eq.cov is TRUE, the permutation method is used to calculate p-values; otherwise, the parametric bootstrap resampling is used.

n.iter

a numeric integer indicating the number of permutation/resampling iterations. The default is 1,000.

cov1.est

This and the following arguments are only effective when eq.cov = FALSE and the parametric bootstrap resampling is used to calculate p-values. This argument specifies a consistent estimate of the covariance matrix of sample population 1 when eq.cov is FALSE. This can be obtained from various apporoaches (e.g., banding, tapering, and thresholding; see Pourahmadi 2013). If not specified, this function uses a banding approach proposed by Bickel and Levina (2008) to estimate the covariance matrix.

cov2.est

a consistent estimate of the covariance matrix of sample population 2 when eq.cov is FALSE. It is similar with the argument cov1.est.

bandwidth1

a vector of nonnegative integers indicating the candidate bandwidths to be used in the banding approach (Bickel and Levina, 2008) for estimating the covariance of sample population 1 when eq.cov is FALSE. This argument is effective when cov1.est is not provided. The default is a vector containing 50 candidate bandwidths chosen from {0, 1, 2, ..., p}.

bandwidth2

similar with the argument bandwidth1; it is used to specify candidate bandwidths for estimating the covariance of sample population 2 when eq.cov is FALSE.

cv.fold

an integer greater than or equal to 2 indicating the fold of cross-validation. The default is 5. See page 211 in Bickel and Levina (2008).

norm

a character string indicating the type of matrix norm for the calculation of risk function in cross-validation. This argument will be passed to the norm function. The default is the Frobenius norm ("F").

seeds

a vector of seeds for each permutation or parametric bootstrap resampling iteration; this is optional.

Details

See the details in apval_Cai2014.

Value

A list including the following elements:

sam.info

the basic information about the two groups of samples, including the samples sizes and dimension.

opt.bw1

the optimal bandwidth determined by the cross-validation when eq.cov was FALSE and cov1.est was not specified.

opt.bw2

the optimal bandwidth determined by the cross-validation when eq.cov was FALSE and cov2.est was not specified.

cov.assumption

the equality assumption on the covariances of the two sample populations; this was specified by the argument eq.cov.

method

this output reminds users that the p-values are obtained using permutation or parametric bootstrap resampling.

pval

the p-value of the test proposed by Cai et al (2014).

Note

This function does not transform the data with their precision matrix (see Cai et al, 2014). To calculate the p-value of the test statisic with transformation, users can input transformed samples to sam1 and sam2.

References

Bickel PJ and Levina E (2008). "Regularized estimation of large covariance matrices." The Annals of Statistics, 36(1), 199–227.

Cai TT, Liu W, and Xia Y (2014). "Two-sample test of high dimensional means under dependence." Journal of the Royal Statistical Society: Series B (Statistical Methodology), 76(2), 349–372.

Pourahmadi M (2013). High-Dimensional Covariance Estimation. John Wiley & Sons, Hoboken, NJ.

See Also

apval_Cai2014

Examples

#library(MASS)
#set.seed(1234)
#n1 <- n2 <- 50
#p <- 200
#mu1 <- rep(0, p)
#mu2 <- mu1
#mu2[1:10] <- 0.2
#true.cov <- 0.4^(abs(outer(1:p, 1:p, "-"))) # AR1 covariance
#sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov)
#sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov)
# increase n.iter to reduce Monte Carlo error
#epval_Cai2014(sam1, sam2, n.iter = 10)

# the two sample populations have different covariances
#true.cov1 <- 0.2^(abs(outer(1:p, 1:p, "-")))
#true.cov2 <- 0.6^(abs(outer(1:p, 1:p, "-")))
#sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov1)
#sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov2)
# increase n.iter to reduce Monte Carlo error
#epval_Cai2014(sam1, sam2, eq.cov = FALSE, n.iter = 10,
#	bandwidth1 = 10, bandwidth2 = 10)