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.
Value
crit
the vector of criterion values.
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.
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).
# NOT RUN {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
# }