Translated to German and added a preamble.

R/T_Statistic_GP.R

+#################################################################################################
+#################################################################################################
+## 
+#################################################################################################
+#################################################################################################
+
+# required packages
 library(Directional)
 library(MASS)
 library(MonteCarlo)
 library(data.table)
 
-setwd("/Users/faik/Desktop/Mathematik/Masterarbeit/Ansatz_2/R-Codes")
+setwd("/home/jaroslav-borodavka/Schreibtisch/Dissertation/Arbeit/Bernoulli Submission/codes/codes_git_project/R")
 
 #################################################################################################
-## Funktionen ## 
+## functions ## 
 
-# Kovarianzkern
+# covariance kernel formulas as per section 2 of the main manuscript
 rho_beta <- function(b, c, beta){
   d = dim(b)[2]
   res = switch (beta,
@@ -22,51 +29,70 @@ rho_beta <- function(b, c, beta){
   return(res)
 }
 
+# uniformly distributed vectors on S^(d-1)
+runif_sphere <- function(n, d){
+  X = mvrnorm(n, rep(0,d), diag(1,d))
+  if (n == 1){
+    temp = as.vector(crossprod(X))
+    U = X/sqrt(temp)
+  }
+  else{
+    temp = diag(X %*% t(X))
+    U = matrix(0, nrow = n, ncol = d)
+    for (j in 1:n){ 
+      U[j,] = X[j,]/sqrt(temp[j])             
+    }
+  }
+  return(U)
+}
+
 T_asymptotic_stat_value <- function(d, beta){
   m = 0
-  if (d<5) m = 3000 else m = 10000
-  runif_cover = rvmf(m, rep(1,d), 0)
-  # Kovarianzmatrix
+  if (d<5) m = 1000 else m = 5000
+  
+  runif_cover = runif_sphere(m, d)
+  
+  # covariance matrix
   Sigma_beta = rho_beta(runif_cover, runif_cover, beta)
   Z_beta = mvrnorm(1, rep(0, m), Sigma_beta)
-  return(list("Realisierung" = max(Z_beta^2)))
+  return(list("realization" = max(Z_beta^2)))
 }
 
 #################################################################################################
-## Berechnung kritischer Werte der Grenzverteilung von T per Monte Carlo ## 
+## calculation of critical values of the asymptotic distribution of T via Monte Carlo ## 
 
-# Umlaeufe
-l = 10000                                                  
-# Parameter
-numCores = 2
+# replication number l (cf. main manuscript, section 6) for critical values
+l = 100000
+# hyperparameters
+numCores = 18
 dim_grid = c(2, 3, 5, 10)
 beta_grid = seq(1, 6)
 grp_number = length(dim_grid)*length(beta_grid)
 
-# Parameterkonstellationen fuer Monte Carlo
+# parameter constellation for MonteCarlo package
 parameter = list("d" = dim_grid, "beta" = beta_grid)  
 
-# Export benötigter Daten, Pakete und Funktionen
+# export of required data, packages and functions
 export_list = list("functions" = "rho_beta", "packages" = c("MASS", "Directional"))
 
-# Monte Carlo Loops
+# Monte Carlo loops
 res_T_asy <- MonteCarlo::MonteCarlo(T_asymptotic_stat_value, nrep = l, param_list = parameter, ncpus = numCores, export_also = export_list)      
-# Erstellung eines Dataframes der Realisierungen
+# producing a data.frame with all realizations
 DF_T_asy <- MonteCarlo::MakeFrame(res_T_asy)                   
 summary(DF_T_asy)
 
-# In data.table umwandeln
+# converting to a data.table
 DT_T_asy <- setDT(DF_T_asy)
 
-# Gruppen bilden
+# creating groups
 DT_T_asy[, grp := .GRP, by = c("d", "beta")]                  
-DT_T_asy = DT_T_asy[order(grp, Realisierung)]
-DT_T_asy = DT_T_asy[, list("Dimension" = d, "Beta" = beta, "Realisierung" = Realisierung, 
-                                             "Quantil_90" = lapply(.SD, quantile, probs = 0.90),
-                                             "Quantil_95" = lapply(.SD, quantile, probs = 0.95),
-                                             "Quantil_99" = lapply(.SD, quantile, probs = 0.99)), 
-                                      by = grp, .SDcols = c("Realisierung")]
+DT_T_asy = DT_T_asy[order(grp, realization)]
+DT_T_asy = DT_T_asy[, list("dimension" = d, "beta" = beta, "realization" = realization, 
+                                             "level_10" = lapply(.SD, quantile, probs = 0.90),
+                                             "level_5" = lapply(.SD, quantile, probs = 0.95),
+                                             "level_1" = lapply(.SD, quantile, probs = 0.99)), 
+                                      by = grp, .SDcols = c("realization")]
 print(DT_T_asy)
 
-save.image(file = "T_Statistic_GP_Daten.Rdata")
+save.image(file = "T_Statistic_GP_data.Rdata")