Translated to German and added a preamble.

R/Comp_Statistics.R

+#################################################################################################
+#################################################################################################
+## This script collects all the competing test statistics that are used in the simulation study,
+## cf. section 6 of the main manuscript. It is not specifically loaded in other scripts, but 
+## might serve useful for some toy example tests.
+#################################################################################################
+#################################################################################################
+
+# required packages
 library(tictoc)
 library(Directional)
 library(MASS)
 library(MonteCarlo)
 library(data.table)
-library(zipfR)        # incomplete Beta function
+library(zipfR)        # contains incomplete Beta function
 
 #################################################################################################
-## Konkurrierende Teststatistiken ## 
+## competing test statistics ## 
 
-# Transformation von Einheitsvektoren in zirkuläre Daten via Bogenmaß (nur d = 2), siehe n-sphere auf Wiki
+# transformation of unit vectors into circular data via radian (only d = 2)
 euclidtoangular <- function(unit_data){
   return(matrix(data = atan2(unit_data[,2], unit_data[,1]), ncol = 1))
 }
 
-# Testentscheid 
+# test decision 
 Decision <- function(teststatistic_value, critical){              
   
   decision <- 0
@@ -26,7 +35,7 @@ Decision <- function(teststatistic_value, critical){
   return(decision)
 }
 
-# Gleichverteilung auf S^(d-1)
+# uniformly distributed vectors on S^(d-1)
 runif_sphere <- function(n, d){
   X = mvrnorm(n, rep(0,d), diag(1,d))
   if (n == 1){
@@ -43,17 +52,17 @@ runif_sphere <- function(n, d){
   return(U)
 }
 
-# Kuiper-Teststatistik basierend auf Directional-Paket (nur d = 2)
+# Kuiper test based on Directional package (only d = 2), data in form of unit vectors
 Kuiper <- function(data){
   return(Directional::kuiper(euclidtoangular(data), rads = TRUE)[1])
 }
 
-# Watson-Teststatistik basierend auf Directional-Paket (nur d = 2)
+# Watson test based on Directional package (only d = 2), data in form of unit vectors
 Watson <- function(data){
   return(Directional::watson(euclidtoangular(data), rads = TRUE)[1])
 }
 
-# Cuesta-Albertos-Teststatistik
+# Cuesta-Albertos tests (d=2 and d>2)
 F_CA <- function(d){
   if (d==2){
     res <- function(t) 1 - acos(t)/pi
@@ -72,7 +81,7 @@ Cuesta_Albertos_unique <- function(data, rdirection){
   return(ks.test(projected_data, "cdf")$p.value)
 }
 
-# 100 gleichverteilte Richtungen fuer die CA-Teststatistik
+# 100 uniformly distributed directions
 rdirections_2 <- runif_sphere(100, 2)
 rdirections_3 <- runif_sphere(100, 3)
 rdirections_5 <- runif_sphere(100, 5)
@@ -93,12 +102,12 @@ Cuesta_Albertos_multi <- function(data){
   return(min(p_values))
 }
 
-# Rayleigh-Teststatistik basierend auf Directional-Paket 
+# Rayleigh test based on Directional package 
 Rayleigh <- function(data){
   return(Directional::rayleigh(data, modif = TRUE)[1])
 }
 
-# Ajne-Teststatistik
+# Ajne test
 Ajne <- function(data){
   n = dim(data)[1]
   temp = data %*% t(data)
@@ -106,7 +115,7 @@ Ajne <- function(data){
   return(n/4 - 1/(n*pi)*sum(Psi))
 }
 
-# Bingham-Teststatistik
+# Bingham test 
 Bingham <- function(data){
   n = dim(data)[1]
   d = dim(data)[2]
@@ -118,7 +127,7 @@ Bingham <- function(data){
   return(n*d*(d+2)/2*(sum(diag(S %*% S)) - 1/d))
 }
 
-# Giné-Teststatistik
+# Giné test
 Gine_G <- function(data){
   n = dim(data)[1]
   d = dim(data)[2]
@@ -127,7 +136,7 @@ Gine_G <- function(data){
   return(n/2 - (d-1)*gamma((d-1)/2)^2/(2*n*gamma(d/2)^2)*sum(sin(Psi)))
 }
 
-# Cramér-von Mises-Teststatistik
+# Cramér-von Mises test
 integrand <- function(t, angle, dim){
   return(F_CA(dim)(t)*F_CA(dim-1)(t*tan(angle/2)/sqrt(1-t^2))/Cbeta(a = 1/2, b = (dim-1)/2)*(1-t^2)^((dim-3)/2))
 }