two sample cvm.test

library(data.table)

cvm.test <- function(x, y) {
  DT <- data.table(value = c(sort(x), sort(y)), id = rep(1:2, c(length(x), length(y))),
                   rank = c(1:length(x), 1:length(y)))
  DT[ , overall_rank := rank(DT$value)]
  statDT <- DT[ , .(U = sum((rank - overall_rank)**2), N = .N), by = .(id)]
  prod_N <- product(statDT$N)
  sum(statDT$N * statDT$U) / (prod_N * nrow(DT)) - (4 * prod_N - 1) / 6 / nrow(DT)
}

mean(sapply(1:1e4, function(i){
  X <- rnorm(40, 4, 7)
  Y <- rnorm(20, 4, 7)
  cvm.test(X, Y) > 0.461
}))
# 0.0513

mean(sapply(1:1e4, function(i){
  X <- rnorm(40, 4, 7)
  Y <- rnorm(20, 4, 10)
  cvm.test(X, Y) > 0.461
}))
# 0.0913

mean(sapply(1:1e4, function(i){
  X <- rnorm(40, 4, 7)
  Y <- rnorm(20, 4, 14)
  cvm.test(X, Y) > 0.461
}))
# 0.2315

mean(sapply(1:1e4, function(i){
  X <- rnorm(40, 4, 7)
  Y <- rnorm(20, 8, 7)
  cvm.test(X, Y) < 0.461
}))
# 0.4832