Package 'subrank' reference manual

Title:	Computes Copula using Ranks and Subsampling
Description:	Estimation of copula using ranks and subsampling. The main feature of this method is that simulation studies show a low sensitivity to dimension, on realistic cases.
Authors:	Jerome Collet
Maintainer:	Jerome Collet <[email protected]>
License:	GPL (>= 3)
Version:	0.9.9.3
Built:	2024-11-18 06:34:54 UTC
Source:	CRAN

Computes Copula using Ranks and Subsampling

Description

Estimation of copula using ranks and subsampling. The main feature of this method is that simulation studies show a low sensitivity to dimension, on realistic cases.

Details

The DESCRIPTION file:

Package:	subrank
Type:	Package
Title:	Computes Copula using Ranks and Subsampling
Version:	0.9.9.3
Date:	2023-04-06
Author:	Jerome Collet
Maintainer:	Jerome Collet <[email protected]>
Description:	Estimation of copula using ranks and subsampling. The main feature of this method is that simulation studies show a low sensitivity to dimension, on realistic cases.
License:	GPL (>= 3)
LazyLoad:	yes
NeedsCompilation:	yes
Packaged:	2023-04-05 13:54:30 UTC; Jerome
Repository:	CRAN
Date/Publication:	2023-04-05 17:50:05 UTC

Index of help topics:

corc                    Function to estimate copula using ranks and
                        sub-sampling
corc0                   Function to estimate copula using ranks and
                        sub-sampling, minimal version.
desscop                 Discrete copula graph, a two-dimensional
                        projection
desscoptous             Discrete copula graph, ALL two-dimensional
                        projections
estimdep                Dependence estimation
predictdep              Probability forecasting
predonfly               Probability forecasting
simany                  Test statistic distribution under any
                        hypothesis
simnul                  Test statistic distribution under independence
                        hypothesis
subrank-package         Computes Copula using Ranks and Subsampling

Taking a sample, its dimension, and a sub-sample size, allows to estimate a discretized copula. This object has interesting features: convergence to copula, robustness with respect to dimension.

Author(s)

Jerome Collet

Maintainer: Jerome Collet <[email protected]>

Examples

lon <- 31
a <- 2.85
x <- rnorm(lon)
y = a*x^2+rnorm(lon)
tablo = as.data.frame(cbind(x,y))
c=corc(tablo,c(1,2),8)
desscop(c,1,2)
lon <- 31
a <- 2.85
x <- rnorm(lon)
y = a*x^2+rnorm(lon)
tablo = as.data.frame(cbind(x,y))
c=corc(tablo,c(1,2),8)
desscop(c,1,2)

Function to estimate copula using ranks and sub-sampling

Description

Takes a sample, its dimension, a sub-sample size, and returns a discrete copula.

Usage

corc(dataframe, varnames, subsampsize, nbsafe=5,mixties=FALSE,nthreads=2)
corc(dataframe, varnames, subsampsize, nbsafe=5,mixties=FALSE,nthreads=2)

Arguments

`dataframe`	a data frame, containing the observations
`varnames`	the name of the variables we want to estimate the dependence between
`subsampsize`	the sub-sample size
`nbsafe`	the ratio between the number of sub-samples and the cardinality of the discretized copula.
`mixties`	if `TRUE`, put equal weight on tied values, using random permutations
`nthreads`	number of number of threads, assumed to be strictly positive. For "full throttle" computations, consider using parallel::detectCores()

Value

`cop`	an array representing the discretized copula
`ties`	the number of sub-samples with a tie
`nsubsampreal`	the effective number of sub-samples drawn
`varnames`	the name of the variables studied
`nnm`	the number of observations without missing values

Author(s)

Jerome Collet

Examples

lon <- 30
a <- 2
x <- rnorm(lon)
y = a*x^2+rnorm(lon)
datatable = as.data.frame(cbind(x,y))
c=corc(datatable,c("x","y"),8)
c
sum(c$cop)
lon <- 30
a <- 2
x <- rnorm(lon)
y = a*x^2+rnorm(lon)
datatable = as.data.frame(cbind(x,y))
c=corc(datatable,c("x","y"),8)
c
sum(c$cop)

Function to estimate copula using ranks and sub-sampling, minimal version.

Description

Minimal version of function corc.

Usage

corc0(datavector,sampsize,dimension,subsampsize,nboot,u,mixties=FALSE,nthreads=2)
corc0(datavector,sampsize,dimension,subsampsize,nboot,u,mixties=FALSE,nthreads=2)

Arguments

`datavector`	a vector, containing the observations
`sampsize`	the sample size
`dimension`	the sample dimension
`subsampsize`	the sub-sample size
`nboot`	the number of sub-samples (must be big)
`u`	a random seed, integer
`mixties`	if `TRUE`, put equal weight on tied values, using random permutations
`nthreads`	number of number of threads, assumed to be strictly positive. For "full throttle" computations, consider using parallel::detectCores()

Value

the number of hits for each vector of ranks, plus 2 last values of the vector : number of ties and number of sub-samples really used.

Author(s)

Jerome Collet

Examples

lon <- 30
a <- 2.85
x <- rnorm(lon)
y = a*x^2+rnorm(lon)
c=corc0(c(x,y),lon,2,8,1e5,75014)
c

c0=c(
1203, 1671, 1766, 959, 1586, 1715, 1803, 1205, 1260,1988, 2348, 1917, 3506, 2045, 1340,
1093, 2694, 2757,2233, 1085, 2322, 1793, 1569, 1263, 1709, 1747, 1512,1308, 1778, 1354,
1184, 1097, 2487, 2730, 2112, 1100,2435, 2033, 1572, 1093, 1369, 1722, 1462, 1015, 1228,
1419, 1776, 1852, 1009, 1097, 1179, 1323, 1595, 1316,1477, 2628, 889, 1178, 1981, 4000, 
35, 840, 2091, 4467,0, 27405)
set.seed(75013)
lon=30
dimension=3
sssize=4
c0==corc0(rnorm(lon*dimension),lon,dimension,sssize,1e5,75014)
lon <- 30
a <- 2.85
x <- rnorm(lon)
y = a*x^2+rnorm(lon)
c=corc0(c(x,y),lon,2,8,1e5,75014)
c

c0=c(
1203, 1671, 1766, 959, 1586, 1715, 1803, 1205, 1260,1988, 2348, 1917, 3506, 2045, 1340,
1093, 2694, 2757,2233, 1085, 2322, 1793, 1569, 1263, 1709, 1747, 1512,1308, 1778, 1354,
1184, 1097, 2487, 2730, 2112, 1100,2435, 2033, 1572, 1093, 1369, 1722, 1462, 1015, 1228,
1419, 1776, 1852, 1009, 1097, 1179, 1323, 1595, 1316,1477, 2628, 889, 1178, 1981, 4000, 
35, 840, 2091, 4467,0, 27405)
set.seed(75013)
lon=30
dimension=3
sssize=4
c0==corc0(rnorm(lon*dimension),lon,dimension,sssize,1e5,75014)

Discrete copula graph, a two-dimensional projection

Description

Draws a discrete joint probability, for 2 variables, using bubbles

Usage

desscop(copest, xname, yname, normalize = FALSE, axes = TRUE)
desscop(copest, xname, yname, normalize = FALSE, axes = TRUE)

Arguments

`copest`	the estimated copula (the whole structure resulting from `corc`)
`xname`	the name of the variable we want to put on the horizontal axis
`yname`	the name of the variable we want to put on the vertical axis
`normalize`	if TRUE, the smallest probability is rescaled to 0, and the largest to 1
`axes`	if TRUE, puts the name of the variables on the axes

Author(s)

Jerome Collet

Examples

lon <- 31
a <- 2.85
x <- rnorm(lon)
y = a*x^2+rnorm(lon)
tablo = as.data.frame(cbind(x,y))
c=corc(tablo,c("x","y"),8)
desscop(c,"x","y")

tablo = as.data.frame(cbind(x=rep(0,each=lon),y=rep(0,each=lon)))
c=corc(tablo,c("x","y"),8,mixties=TRUE)
desscop(c,"x","y")

lon <- 31
a <- 2.85
x <- rnorm(lon)
y = a*x^2+rnorm(lon)
tablo = as.data.frame(cbind(x,y))
c=corc(tablo,c("x","y"),8)
desscop(c,"x","y")

tablo = as.data.frame(cbind(x=rep(0,each=lon),y=rep(0,each=lon)))
c=corc(tablo,c("x","y"),8,mixties=TRUE)
desscop(c,"x","y")

Discrete copula graph, ALL two-dimensional projections

Description

Draws a discrete joint probability, for 2 variables, using bubbles

Usage

desscoptous(copest, normalize = FALSE)
desscoptous(copest, normalize = FALSE)

Arguments

`copest`	the estimated copula (the whole structure resulting from `corc`)
`normalize`	if TRUE, the smallest probability is rescaled to 0, and the largest to 1

Author(s)

Jerome Collet

Examples

lon <- 31
a <- 2.85
x <- rnorm(lon)
y = a*x^2+rnorm(lon)
z = rnorm(lon)
tablo = as.data.frame(cbind(x,y,z))
c=corc(tablo,c("x","y","z"),8)
desscoptous(c)
lon <- 31
a <- 2.85
x <- rnorm(lon)
y = a*x^2+rnorm(lon)
z = rnorm(lon)
tablo = as.data.frame(cbind(x,y,z))
c=corc(tablo,c("x","y","z"),8)
desscoptous(c)

Dependence estimation

Description

From a set of observations, builds a description of the multivariate distribution

Usage

estimdep(dataframe,varnames,subsampsize,nbsafe=5,mixties=FALSE,nthreads=2)
estimdep(dataframe,varnames,subsampsize,nbsafe=5,mixties=FALSE,nthreads=2)

Arguments

`dataframe`	a data frame containing the observations
`varnames`	the name of the variables we want to estimate the multivariate distribution
`subsampsize`	the sub-sample size
`nbsafe`	the ratio between the discretized copula size and the number of sub-samples
`mixties`	if `TRUE`, put equal weight on tied values, using random permutations
`nthreads`	number of number of threads, assumed to be strictly positive. For "full throttle" computations, consider using parallel::detectCores()

Value

the description of the dependence, it is an object with the following parts:

`cop`	the array representing the discretized copula
`margins`	the matrix representing the margins, estimated using kernel density estimation
`varnames`	the names of the variables

Author(s)

Jerome Collet

Examples

lon=3000
plon=3000
subsampsize=20

##############
x=(runif(lon)-1/2)*3
y=x^2+rnorm(lon)
z=rnorm(lon)
donori=as.data.frame(cbind(x,y,z))
depori=estimdep(donori,c("x","y","z"),subsampsize)

knownvalues=data.frame(z=rnorm(plon))
prev <- predictdep(knownvalues,depori)
plot(prev$x,prev$y,xlim=c(-2,2),ylim=c(-2,5),pch=20,cex=0.5)
points(donori[,1:2],col='red',pch=20,cex=.5)

knownvalues=data.frame(x=(runif(lon)-1/2)*3)
prev <- predictdep(knownvalues,depori)
plot(prev$x,prev$y,xlim=c(-2,2),ylim=c(-2,5),pch=20,cex=0.5)
points(donori[,1:2],col='red',pch=20,cex=.5)

knownvalues=data.frame(y=runif(plon,min=-2,max=4))
prev <- predictdep(knownvalues,depori)
plot(prev$x,prev$y,xlim=c(-2,2),ylim=c(-2,5),pch=20,cex=0.5)
points(donori[,1:2],col='red',pch=20,cex=.5)
lon=3000
plon=3000
subsampsize=20

##############
x=(runif(lon)-1/2)*3
y=x^2+rnorm(lon)
z=rnorm(lon)
donori=as.data.frame(cbind(x,y,z))
depori=estimdep(donori,c("x","y","z"),subsampsize)

knownvalues=data.frame(z=rnorm(plon))
prev <- predictdep(knownvalues,depori)
plot(prev$x,prev$y,xlim=c(-2,2),ylim=c(-2,5),pch=20,cex=0.5)
points(donori[,1:2],col='red',pch=20,cex=.5)

knownvalues=data.frame(x=(runif(lon)-1/2)*3)
prev <- predictdep(knownvalues,depori)
plot(prev$x,prev$y,xlim=c(-2,2),ylim=c(-2,5),pch=20,cex=0.5)
points(donori[,1:2],col='red',pch=20,cex=.5)

knownvalues=data.frame(y=runif(plon,min=-2,max=4))
prev <- predictdep(knownvalues,depori)
plot(prev$x,prev$y,xlim=c(-2,2),ylim=c(-2,5),pch=20,cex=0.5)
points(donori[,1:2],col='red',pch=20,cex=.5)

From a set of incomplete observations, and a description of the dependence, provides simulated values of the unknown coordinates. It is also possible to simulate unconditionally, with empty observations.

Usage

predictdep(knownvalues,dependence,smoothing=c("Uniform","Beta"),nthreads=2)
predictdep(knownvalues,dependence,smoothing=c("Uniform","Beta"),nthreads=2)

Arguments

`knownvalues`	in case of conditional simulation, a matrix containing incomplete observations, the known coordinates being the same for all observations. If no variable name in `knwonvalues` appears in `dependence$varnames`, then the simulation is unconditional.
`dependence`	the description of the dependence we want to use to forecast, as built by function `estimdep`
`smoothing`	the smoothing method for input and output ranks.
`nthreads`	number of number of threads, assumed to be strictly positive. For "full throttle" computations, consider using parallel::detectCores()

Value

the matrix of the completed observations

Author(s)

Jerome Collet

Examples

lon=100
plon=100
subsampsize=10

shift=0
noise=0
knowndims=1

x=rnorm(lon)
y=2*x+noise*rnorm(lon)
donori=as.data.frame(cbind(x,y))
depori=estimdep(donori,c("x","y"),subsampsize)
##
knownvalues=data.frame(x=rnorm(plon)+shift)
prev <- predictdep(knownvalues,depori)
##
plot(prev$x,prev$y,xlim=c(-2,2),ylim=c(-2,5),pch=20,cex=0.5)
points(donori[,1:2],col='red',pch=20,cex=.5)
##
knownvalues=data.frame(x=rnorm(plon)+shift)
prev <- predictdep(knownvalues,depori,smoothing="Beta")
##
plot(prev$x,prev$y,xlim=c(-2,2),ylim=c(-2,5),pch=20,cex=0.5)
points(donori[,1:2],col='red',pch=20,cex=.5)

# souci normal si |shift|>>1

knownvalues=data.frame(z=rnorm(plon)+shift)
prev <- predictdep(knownvalues,depori)
##
plot(prev$x,prev$y,xlim=c(-2,2),ylim=c(-2,5),pch=20,cex=0.5)
points(donori[,1:2],col='red',pch=20,cex=.5)
lon=100
plon=100
subsampsize=10

shift=0
noise=0
knowndims=1

x=rnorm(lon)
y=2*x+noise*rnorm(lon)
donori=as.data.frame(cbind(x,y))
depori=estimdep(donori,c("x","y"),subsampsize)
##
knownvalues=data.frame(x=rnorm(plon)+shift)
prev <- predictdep(knownvalues,depori)
##
plot(prev$x,prev$y,xlim=c(-2,2),ylim=c(-2,5),pch=20,cex=0.5)
points(donori[,1:2],col='red',pch=20,cex=.5)
##
knownvalues=data.frame(x=rnorm(plon)+shift)
prev <- predictdep(knownvalues,depori,smoothing="Beta")
##
plot(prev$x,prev$y,xlim=c(-2,2),ylim=c(-2,5),pch=20,cex=0.5)
points(donori[,1:2],col='red',pch=20,cex=.5)

# souci normal si |shift|>>1

knownvalues=data.frame(z=rnorm(plon)+shift)
prev <- predictdep(knownvalues,depori)
##
plot(prev$x,prev$y,xlim=c(-2,2),ylim=c(-2,5),pch=20,cex=0.5)
points(donori[,1:2],col='red',pch=20,cex=.5)

Probability forecasting

Description

From two sets of observations, first one of complete observations and second one of incomplete observations, provides simulated values of the unknown coordinates.

Usage

predonfly(completeobs,incompleteobs,varnames,subsampsize,nbpreds=1,mixties=FALSE,
          maxtirs=1e5,complete=TRUE,nthreads=2)
predonfly(completeobs,incompleteobs,varnames,subsampsize,nbpreds=1,mixties=FALSE,
          maxtirs=1e5,complete=TRUE,nthreads=2)

Arguments

`completeobs`	the set of complete observations.
`incompleteobs`	the set of incomplete observations.
`varnames`	the modeled variables.
`subsampsize`	the sub-sample size.
`nbpreds`	the number of predictions for each incomplete observation.
`mixties`	if `TRUE`, should put equal weight on tied values, using random permutations (not yet implemented)
`maxtirs`	the maximum number of sub-samples, to stop the computation even if they did not provide `nbpreds` predictions for each incomplete observation.
`complete`	If `TRUE`, predictions are completed with incomplete observations
`nthreads`	number of number of threads, assumed to be strictly positive. For "full throttle" computations, consider using parallel::detectCores()

Value

the matrix of the completed observations

Author(s)

Jerome Collet

Examples

lon=100
plon=30
subsampsize=10

x=rnorm(lon)
y=2*x+rnorm(lon)*0
donori=as.data.frame(cbind(x,y))
##
knownvalues=data.frame(x=rnorm(plon))
prev <- predonfly(donori,knownvalues,c("x","y"),subsampsize,100)

##
plot(prev$x,prev$y,pch=20,cex=0.5,
     ylim=range(c(prev$y,donori$y),na.rm=TRUE),xlim=range(c(prev$x,donori$x)))
points(donori[,1:2],col='red',pch=20,cex=.5)

lon=3000
mg=20
dimtot=4
rayon=6

genboules <- function(lon,a,d)
{
  ss <- function(vec)
  {return(sum(vec*vec))}
  surface=matrix(nrow=lon,ncol=d,data=rnorm(lon*d))
  rayons=sqrt(apply(surface,1,ss))
  surface=surface/rayons
  return(matrix(nrow=lon,ncol=d,data=rnorm(lon*d))+a*surface)
}

##############

donori=genboules(lon,rayon,dimtot)
donori=as.data.frame(donori)

dimconnues=3:dimtot
valconnues=matrix(nrow=1,ncol=length(dimconnues),data=0)
valconnues=as.data.frame(valconnues)
names(valconnues)=names(donori)[3:dimtot]
prev <- predonfly(donori,valconnues,names(donori),subsampsize,100)

boule2=genboules(plon,rayon,2)

plot(boule2[,1:2],xlab='X1',ylab='X2',pch=20,cex=.5)
plot(prev$V1,prev$V2,xlab='X1',ylab='X2',pch=20,cex=.5)
lon=100
plon=30
subsampsize=10

x=rnorm(lon)
y=2*x+rnorm(lon)*0
donori=as.data.frame(cbind(x,y))
##
knownvalues=data.frame(x=rnorm(plon))
prev <- predonfly(donori,knownvalues,c("x","y"),subsampsize,100)

##
plot(prev$x,prev$y,pch=20,cex=0.5,
     ylim=range(c(prev$y,donori$y),na.rm=TRUE),xlim=range(c(prev$x,donori$x)))
points(donori[,1:2],col='red',pch=20,cex=.5)

lon=3000
mg=20
dimtot=4
rayon=6

genboules <- function(lon,a,d)
{
  ss <- function(vec)
  {return(sum(vec*vec))}
  surface=matrix(nrow=lon,ncol=d,data=rnorm(lon*d))
  rayons=sqrt(apply(surface,1,ss))
  surface=surface/rayons
  return(matrix(nrow=lon,ncol=d,data=rnorm(lon*d))+a*surface)
}

##############

donori=genboules(lon,rayon,dimtot)
donori=as.data.frame(donori)

dimconnues=3:dimtot
valconnues=matrix(nrow=1,ncol=length(dimconnues),data=0)
valconnues=as.data.frame(valconnues)
names(valconnues)=names(donori)[3:dimtot]
prev <- predonfly(donori,valconnues,names(donori),subsampsize,100)

boule2=genboules(plon,rayon,2)

plot(boule2[,1:2],xlab='X1',ylab='X2',pch=20,cex=.5)
plot(prev$V1,prev$V2,xlab='X1',ylab='X2',pch=20,cex=.5)

Test statistic distribution under any hypothesis

Description

Simulates the test statistic, under independence

Usage

simany(sampsize,dimension,subsampsizes,sampnum,nbsafe=5,nthreads=2, fun=NULL, ...)
simany(sampsize,dimension,subsampsizes,sampnum,nbsafe=5,nthreads=2, fun=NULL, ...)

Arguments

`sampsize`	sample size
`dimension`	sample dimension
`subsampsizes`	vector of sub-sample sizes
`sampnum`	number of samples
`nbsafe`	the ratio between the number of sub-samples and the cardinality of the discretized copula.
`nthreads`	number of number of threads, assumed to be strictly positive. For "full throttle" computations, consider using parallel::detectCores()
`fun`	the function describing the dependence.
`...`	optional arguments to `fun`

Value

`lrs`	the distances with independent case
`lrs2mean`	the distances with theoretical value, given dependence `fun`
`scarcities`	the proportions of non-reached vector ranks
`DistTypes`	a recall of the list of the distance types: "KL","L2","L1","APE"

Author(s)

Jerome Collet

Examples

depquad <- function(lon,dd,a)
{
  x <- rnorm(lon)
  y0 <- a*x^2
  y <- y0 + rnorm(lon)
  reste=rnorm((dd-2)*lon)
  return(c(x,y,reste))
}
sims0=simany(101,3,8,50,nbsafe=1)
seuils=apply(sims0$lrs,3,quantile,0.95)
seuils=matrix(ncol=4,nrow=50,seuils,byrow=TRUE)
sims1=simany(101,3,8,50,nbsafe=1,fun=depquad,a=0.5)
apply(sims1$lrs[,1,]>seuils,2,mean)
depquad <- function(lon,dd,a)
{
  x <- rnorm(lon)
  y0 <- a*x^2
  y <- y0 + rnorm(lon)
  reste=rnorm((dd-2)*lon)
  return(c(x,y,reste))
}
sims0=simany(101,3,8,50,nbsafe=1)
seuils=apply(sims0$lrs,3,quantile,0.95)
seuils=matrix(ncol=4,nrow=50,seuils,byrow=TRUE)
sims1=simany(101,3,8,50,nbsafe=1,fun=depquad,a=0.5)
apply(sims1$lrs[,1,]>seuils,2,mean)

Test statistic distribution under independence hypothesis

Description

Simulates the test statistic, under independence

Usage

simnul(sampsize, dimension, subsampsizes, sampnum,KL=TRUE,nbsafe=5,nthreads=2)
simnul(sampsize, dimension, subsampsizes, sampnum,KL=TRUE,nbsafe=5,nthreads=2)

Arguments

`sampsize`	sample size
`dimension`	sample dimension
`subsampsizes`	vector of sub-sample sizes
`sampnum`	number of samples
`KL`	if TRUE, returns the Kullback-Leibler divergence with the independent case, if FALSE, the L2 distance. There is no re-normalization, contrary to what happens for `simany`.
`nbsafe`	the ratio between the number of sub-samples and the cardinality of the discretized copula.
`nthreads`	number of number of threads, assumed to be strictly positive. For "full throttle" computations, consider using parallel::detectCores()

Value

`lrs`	the distances with independent case
`scarcities`	the proportions of non-reached vector ranks

Author(s)

Jerome Collet

Examples

library(datasets)
# plot(swiss)
c=corc(swiss,1:3,8)
c
RV=sum(c$cop*log(c$cop),na.rm=TRUE)+3*log(8)
sims=simnul(47,3,8,100)
pvalue=mean(RV<sims$lrs)
pvalue
RV
summary(sims$lrs)
library(datasets)
# plot(swiss)
c=corc(swiss,1:3,8)
c
RV=sum(c$cop*log(c$cop),na.rm=TRUE)+3*log(8)
sims=simnul(47,3,8,100)
pvalue=mean(RV<sims$lrs)
pvalue
RV
summary(sims$lrs)

Package 'subrank'

Help Index

Computes Copula using Ranks and Subsampling

Description

Details

Author(s)

Examples

Function to estimate copula using ranks and sub-sampling

Description

Usage

Arguments

Value

Author(s)

Examples

Function to estimate copula using ranks and sub-sampling, minimal version.

Description

Usage

Arguments

Value

Author(s)

Examples

Discrete copula graph, a two-dimensional projection

Description

Usage

Arguments

Author(s)

Examples

Discrete copula graph, ALL two-dimensional projections

Description

Usage

Arguments

Author(s)

Examples

Dependence estimation

Description

Usage

Arguments

Value

Author(s)

Examples

Probability forecasting

Description

Usage

Arguments

Value

Author(s)

Examples

Probability forecasting

Description

Usage

Arguments

Value

Author(s)

Examples

Test statistic distribution under any hypothesis

Description

Usage

Arguments

Value

Author(s)

Examples

Test statistic distribution under independence hypothesis

Description

Usage

Arguments

Value

Author(s)

Examples