Package 'ConvergenceConcepts' reference manual

Title:	Seeing Convergence Concepts in Action
Description:	This is a pedagogical package, designed to help students understanding convergence of random variables. It provides a way to investigate interactively various modes of convergence (in probability, almost surely, in law and in mean) of a sequence of i.i.d. random variables. Visualisation of simulated sample paths is possible through interactive plots. The approach is illustrated by examples and exercises through the function 'investigate', as described in Lafaye de Micheaux and Liquet (2009) <doi:10.1198/tas.2009.0032>. The user can study his/her own sequences of random variables.
Authors:	Pierre Lafaye De Micheaux [aut, cre], Benoit Liquet [aut]
Maintainer:	Pierre Lafaye De Micheaux <[email protected]>
License:	GPL (>= 2)
Version:	1.2.3
Built:	2024-12-03 06:32:43 UTC
Source:	CRAN

Check convergence

Description

This function enables one to investigate the four classical modes of convergence on simulated data: in probability, almost surely, in r-th mean and in law.

Usage

check.convergence(nmax,M,genXn,argsXn=NULL,mode="p",epsilon=0.05,r=2,nb.sp=10,
  density=FALSE,densfunc=dnorm,probfunc=pnorm,tinf=-3,tsup=3,plotfunc=plot,...)
check.convergence(nmax,M,genXn,argsXn=NULL,mode="p",epsilon=0.05,r=2,nb.sp=10,
  density=FALSE,densfunc=dnorm,probfunc=pnorm,tinf=-3,tsup=3,plotfunc=plot,...)

Arguments

`nmax`	number of points in each sample path.
`M`	number of sample paths to be generated.
`genXn`	a function that generates the Xn-X values, or only the Xn values in the law case.
`argsXn`	a list of arguments to `genXn`.
`mode`	a character string specifying the mode of convergence to be investigated, must be one of "p" (default), "as", "r" or "L".
`epsilon`	a numeric value giving the interval endpoint.
`r`	a numeric value (r>0) if convergence in r-th mean is to be studied.
`nb.sp`	number of sample paths to be drawn on the left plot.
`density`	if density=TRUE, then the plot of the density of X and the histogram of Xn is returned. If density=FALSE, then the plot of the distribution function F(t) of X and the empirical distribution Fn(t) of Xn is returned.
`densfunc`	function to compute the density of X.
`probfunc`	function to compute the distribution function of X.
`tinf`	lower limit for investigating convergence in law.
`tsup`	upper limit for investigating convergence in law.
`plotfunc`	R function used to draw the plot: for example `plot` or `points`.
`...`	optional arguments to `plotfunc`.

Details

The objective of this function is to investigate graphically the convergence of some random variable Xn to some random variable X. In order to use it, you should be able to provide generators of Xn and X (or of Xn-X). The four modes of convergence that you can try are: in probability, almost surely, in r-th mean and in law. For the convergence in law, we compute $\hat(l)_n(t)=|\hat{F}_n(t)-F(t)|$ for ten values equally distributed between tinf and tsup.

Author(s)

P. Lafaye de Micheaux and B. Liquet

References

Lafaye de Micheaux, P. ([email protected]), Liquet, B. "Understanding Convergence Concepts: a Visual-Minded and Graphical Simulation-Based Approach", The American Statistician, 63:2, 173–178, (2009).

Examples


## Not run: 

####################### Exercise 3 ##############################
# Let X1, X2, ..., Xn be independent random variables such that #
# P[Xn=sqrt(n)]=1/n and P[Xn=0]=1-1/n                           #
# Does Xn converges to 0 in 2-th mean? in probability?          #
#################################################################

       options(example.ask=FALSE)

       pnotrgen<-function(n){rbinom(n,1,1/(1:n))*sqrt(1:n)}

       check.convergence(nmax=1000,M=500,genXn=pnotrgen,mode="r",r=2)
       legend(100,6,legend=expression(hat(e)["n,2"]),lty=1)
       tt3.1 <<- check.convergence(nmax=1000,M=500,genXn=pnotrgen,mode="p")


## End(Not run)

## Not run: 

####################### Exercise 3 ##############################
# Let X1, X2, ..., Xn be independent random variables such that #
# P[Xn=sqrt(n)]=1/n and P[Xn=0]=1-1/n                           #
# Does Xn converges to 0 in 2-th mean? in probability?          #
#################################################################

       options(example.ask=FALSE)

       pnotrgen<-function(n){rbinom(n,1,1/(1:n))*sqrt(1:n)}

       check.convergence(nmax=1000,M=500,genXn=pnotrgen,mode="r",r=2)
       legend(100,6,legend=expression(hat(e)["n,2"]),lty=1)
       tt3.1 <<- check.convergence(nmax=1000,M=500,genXn=pnotrgen,mode="p")


## End(Not run)

Convergence criterion computation

Description

This function computes the values of the criterion convergence function for convergence in probability, almost surely or in r-th mean, given the sample paths.

Usage

criterion(data,epsilon=0.05,mode="p",r=2)
criterion(data,epsilon=0.05,mode="p",r=2)

Arguments

`data`	matrix containing the sample paths of Xn-X values.
`epsilon`	a numeric value giving the interval endpoint.
`mode`	a character string specifying the mode of convergence to be investigated, must be one of "p" (default), "as" or "r".
`r`	a numeric value (r>0) if convergence in r-th mean is to be studied.

Details

The data matrix contains the $X_n-X$ values. If mode="p", criterion approximates $p_n=P[|X_n-X|>\epsilon]$ . If mode="as", criterion approximates $a_n=P[\exists k\geq n;|X_k-X|>\epsilon]$ . If mode="r", criterion approximates $e_{n,r}=E|X_n-X|^r$ . The approximations are based on the frequentist approach.

Value

crit

the vector of criterion values.

Author(s)

P. Lafaye de Micheaux and B. Liquet

References

Lafaye de Micheaux, P. ([email protected]), Liquet, B. "Understanding Convergence Concepts: a Visual-Minded and Graphical Simulation Based Approach", The American Statistician, 63:2, 173–178, (2009).

Examples

myrbinom <- function(n,alpha){rbinom(n,1,1/(1:n))*((1:n)**alpha)}
data <- generate(nmax=1000,M=500,myrbinom,args=list(alpha=0.5))$data
critr1 <- criterion(data,mode="r",r=1)$crit
myrbinom <- function(n,alpha){rbinom(n,1,1/(1:n))*((1:n)**alpha)}
data <- generate(nmax=1000,M=500,myrbinom,args=list(alpha=0.5))$data
critr1 <- criterion(data,mode="r",r=1)$crit

Generation of sample paths

Description

This function generates the sample paths of a sequence of random variables.

Usage

generate(randomgen,nmax=1000,M=500,argsgen=NULL)
generate(randomgen,nmax=1000,M=500,argsgen=NULL)

Arguments

`nmax`	number of points in each sample path.
`M`	number of sample paths to be generated.
`randomgen`	a function that generates the Xn-X values.
`argsgen`	a list of arguments to `randomgen`.

Value

data

matrix containing in each row a sample path of Xn-X values.

Author(s)

P. Lafaye de Micheaux and B. Liquet

References

Examples

myrbinom<-function(n,alpha){rbinom(n,1,1/(1:n))*((1:n)**alpha)}
data <- generate(randomgen=myrbinom,argsgen=list(alpha=0.5))$data
myrbinom<-function(n,alpha){rbinom(n,1,1/(1:n))*((1:n)**alpha)}
data <- generate(randomgen=myrbinom,argsgen=list(alpha=0.5))$data

Investigate examples and exercises from the paper

Description

This function investigates the convergence for all the examples and exercises from the article cited in references.

Usage

investigate()
investigate()

Author(s)

P. Lafaye de Micheaux and B. Liquet

References

Examples

investigate()
investigate()

Interactive 2D convergence in law

Description

This function enables one to draw the interactive 2D plot to investigate convergence in law.

Usage

law.plot2d(data,density=FALSE,densfunc=dnorm,probfunc=pnorm,tinf=-5,tsup=5)
law.plot2d(data,density=FALSE,densfunc=dnorm,probfunc=pnorm,tinf=-5,tsup=5)

Arguments

`data`	matrix containing the sample paths of Xn values.
`density`	if density=TRUE, then the plot of the density of X and the histogram of Xn is returned. If density=FALSE, then the plot of the distribution function F(t) of X and the empirical distribution Fn(t) of Xn is returned.
`densfunc`	function to compute the density of X.
`probfunc`	function to compute the distribution function of X.
`tinf`	lower limit for investigating convergence in law.
`tsup`	upper limit for investigating convergence in law.

Author(s)

P. Lafaye de Micheaux and B. Liquet

References

Examples

rand <- function(n){(cumsum(rchisq(n,df=1))-(1:n))/sqrt(2*(1:n))}
data <- generate(randomgen=rand,nmax=1000,M=500)$data
law.plot2d(data)
rand <- function(n){(cumsum(rchisq(n,df=1))-(1:n))/sqrt(2*(1:n))}
data <- generate(randomgen=rand,nmax=1000,M=500)$data
law.plot2d(data)

Static 3D convergence in law

Description

This function enables one to draw the static 3D plot to investigate convergence in law.

Usage

law.plot3d(data,probfunc,tinf=-5,tsup=5)
law.plot3d(data,probfunc,tinf=-5,tsup=5)

Arguments

`data`	matrix containing the sample paths of Xn values.
`probfunc`	function to compute the distribution function of X.
`tinf`	lower limit for investigating convergence in law.
`tsup`	upper limit for investigating convergence in law.

Author(s)

P. Lafaye de Micheaux and B. Liquet

References

Examples

rand <- function(n){(cumsum(rchisq(n,df=1))-(1:n))/sqrt(2*(1:n))}
data <- generate(randomgen=rand,nmax=300,M=2000)$data
law.plot3d(data,pnorm)
rand <- function(n){(cumsum(rchisq(n,df=1))-(1:n))/sqrt(2*(1:n))}
data <- generate(randomgen=rand,nmax=300,M=2000)$data
law.plot3d(data,pnorm)

Interactive plot for convergence in probability and almost surely

Description

This function enables one to draw the interactive plots used to study convergence in probability and almost surely.

Usage

p.as.plot(data,critp,critas,epsilon=0.05,nb.sp=10,mode="p")
p.as.plot(data,critp,critas,epsilon=0.05,nb.sp=10,mode="p")

Arguments

`data`	matrix containing the sample paths of Xn-X values.
`critp`	vector containing the criterion values for convergence in probability.
`critas`	vector containing the criterion values for convergence almost surely.
`epsilon`	a numeric value giving the interval endpoint.
`nb.sp`	number of sample paths to be drawn on the left plot.
`mode`	a character string specifying the mode of convergence to investigate, must be one of "p" (default), "as" or "r".

Author(s)

P. Lafaye de Micheaux and B. Liquet

References

Examples

myrbinom <- function(n,alpha){rbinom(n,1,1/(1:n))*((1:n)**alpha)}
data <- generate(randomgen=myrbinom,nmax=1000,M=500,argsgen=list(alpha=0.5))$data
critp <- criterion(data,epsilon=0.05,"p")$crit
critas <- criterion(data,epsilon=0.05,"as")$crit
p.as.plot(data,critp,critas,epsilon=0.05,nb.sp=10,mode="p")
myrbinom <- function(n,alpha){rbinom(n,1,1/(1:n))*((1:n)**alpha)}
data <- generate(randomgen=myrbinom,nmax=1000,M=500,argsgen=list(alpha=0.5))$data
critp <- criterion(data,epsilon=0.05,"p")$crit
critas <- criterion(data,epsilon=0.05,"as")$crit
p.as.plot(data,critp,critas,epsilon=0.05,nb.sp=10,mode="p")

Visualisation of convergence criterion curves

Description

This function enables one to visualize the convergence criterion curve.

Usage

visualize.crit(crit,plotfunc=plot,...)
visualize.crit(crit,plotfunc=plot,...)

Arguments

`crit`	vector containing the criterion values.
`plotfunc`	R function used to draw the plot: for example `plot` or `points`.
`...`	optional arguments to `plotfunc`.

Author(s)

P. Lafaye de Micheaux and B. Liquet

References

Examples

myrbinom <- function(n,alpha){rbinom(n,1,1/(1:n))*((1:n)**alpha)}
data <- generate(randomgen=myrbinom,nmax=1000,M=500,argsgen=list(alpha=0.5))$data
critr1 <- criterion(data,epsilon=0.05,mode="r",r=2)$crit
visualize.crit(critr1,main="Convergence in 2-th mean?",col="red")
legend(0,5,legend=expression(hat(e)["n,2"]),lty=1,col="red")
myrbinom <- function(n,alpha){rbinom(n,1,1/(1:n))*((1:n)**alpha)}
data <- generate(randomgen=myrbinom,nmax=1000,M=500,argsgen=list(alpha=0.5))$data
critr1 <- criterion(data,epsilon=0.05,mode="r",r=2)$crit
visualize.crit(critr1,main="Convergence in 2-th mean?",col="red")
legend(0,5,legend=expression(hat(e)["n,2"]),lty=1,col="red")

Visualisation of sample paths

Description

This function enables one to visualize the sample paths.

Usage

visualize.sp(data,epsilon=0.05,nb.sp=10,plotfunc=plot,...)
visualize.sp(data,epsilon=0.05,nb.sp=10,plotfunc=plot,...)

Arguments

`data`	matrix containing the sample paths of Xn-X values.
`epsilon`	a numeric value giving the interval endpoint.
`nb.sp`	number of sample paths to draw on the plot.
`plotfunc`	R function used to draw the plot: for example `plot` or `points`.
`...`	optional arguments to `plotfunc`.

Author(s)

P. Lafaye de Micheaux and B. Liquet

References

Examples

myrand <- function(n,...){cumsum(rnorm(n,...))/(1:n)}
data <- generate(randomgen=myrand,nmax=1000,M=500)$data
visualize.sp(data,main="Ten sample paths",col="grey")
myrand <- function(n,...){cumsum(rnorm(n,...))/(1:n)}
data <- generate(randomgen=myrand,nmax=1000,M=500)$data
visualize.sp(data,main="Ten sample paths",col="grey")

Package 'ConvergenceConcepts'

Help Index

Check convergence

Description

Usage

Arguments

Details

Author(s)

References

See Also

Examples

Convergence criterion computation

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Generation of sample paths

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Investigate examples and exercises from the paper

Description

Usage

Author(s)

References

See Also

Examples

Interactive 2D convergence in law

Description

Usage

Arguments

Author(s)

References

See Also

Examples

Static 3D convergence in law

Description

Usage

Arguments

Author(s)

References

See Also

Examples

Interactive plot for convergence in probability and almost surely

Description

Usage

Arguments

Author(s)

References

See Also

Examples

Visualisation of convergence criterion curves

Description

Usage

Arguments

Author(s)

References

See Also

Examples

Visualisation of sample paths

Description

Usage

Arguments

Author(s)

References

See Also

Examples