criterion: Convergence criterion computation

Description Usage Arguments Details Value Author(s) References See Also Examples

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

1
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|>ε]. If mode="as", criterion approximates a_n=P[it exists k>= n;|X_k-X|>ε]. 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. (plafaye@club.fr), Liquet, B. "Understanding Convergence Concepts: a Visual-Minded and Graphical Simulation Based Approach", The American Statistician , submitted.

See Also

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

Examples

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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


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