# Empirical Permutation- or Resampling-Based p-value of the Test Proposed by Cai et al (2014) 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. ```r 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`. ## Returns 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). ## 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. ## 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`. ## See Also `apval_Cai2014` ## Examples ```r #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) ```

epval_Cai2014 function