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.
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. (plafaye@club.fr), Liquet, B. "Understanding Convergence Concepts: a Visual-Minded and Graphical Simulation-Based Approach", The American Statistician , submitted.
## 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)