criterion {ConvergenceConcepts}

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

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. ([e-mail]), 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