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,...)

Arguments

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. (plafaye@club.fr), Liquet, B. "Understanding Convergence Concepts: a Visual-Minded and Graphical Simulation-Based Approach", The American Statistician, 63:2, 173–178, (2009).

See Also

criterion, generate, investigate, law.plot2d, law.plot3d, p.as.plot, visualize.crit, visualize.sp

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)

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)

Arguments

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

Author(s)

P. Lafaye de Micheaux and B. Liquet

References

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

See Also

check.convergence, generate, investigate, law.plot2d, law.plot3d, p.as.plot, visualize.crit, visualize.sp

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

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)

Arguments

Value

Author(s)

P. Lafaye de Micheaux and B. Liquet

References

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

See Also

check.convergence, criterion, investigate, law.plot2d, law.plot3d, p.as.plot, visualize.crit, visualize.sp

Examples

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()

Author(s)

P. Lafaye de Micheaux and B. Liquet

References

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

See Also

check.convergence, criterion, generate, law.plot2d, law.plot3d, p.as.plot, visualize.crit, visualize.sp

Examples

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)

Arguments

Author(s)

P. Lafaye de Micheaux and B. Liquet

References

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

See Also

check.convergence, criterion, generate, investigate, law.plot3d, p.as.plot, visualize.crit, visualize.sp

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)

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)

Arguments

Author(s)

P. Lafaye de Micheaux and B. Liquet

References

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

See Also

check.convergence, criterion, generate, investigate, law.plot2d, p.as.plot, visualize.crit, visualize.sp

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)

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")

Arguments

Author(s)

P. Lafaye de Micheaux and B. Liquet

References

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

See Also

check.convergence, criterion, generate, investigate, law.plot2d, law.plot3d, visualize.crit, visualize.sp

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")

Visualisation of convergence criterion curves

Description

This function enables one to visualize the convergence criterion curve.

Usage

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

Arguments

Author(s)

P. Lafaye de Micheaux and B. Liquet

References

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

See Also

check.convergence, criterion, generate, investigate, law.plot2d, law.plot3d, p.as.plot, visualize.sp

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")

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,...)

Arguments

Author(s)

P. Lafaye de Micheaux and B. Liquet

References

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

See Also

check.convergence, criterion, generate, investigate, law.plot2d, law.plot3d, p.as.plot, visualize.crit

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")