epval_Cai2014: Empirical Permutation- or Resampling-Based p-value of the...
In highmean: Two-Sample Tests for High-Dimensional Mean Vectors

Description Usage Arguments Details Value Note References See Also Examples

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.

1 2	epval_Cai2014(sam1, sam2, eq.cov = TRUE, n.iter = 1000, cov1.est, cov2.est, bandwidth1, bandwidth2, cv.fold = 5, norm = "F", seeds)

`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.

See the details in apval_Cai2014.

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).

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.

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.

apval_Cai2014

#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)