[R] Counterintuitive Simulation Results

[Spencer Graves]

	  I wonder if someone can help me understand some counterintuitive 
simulation results.  Below please find 12 lines of R code that 
theoretically, to the best of my understanding, should produce 
essentially a flat line with no discernable pattern.  Instead, I see an 
initial dramatic drop followed by a slow rise to an asymptote.

	  The simulation computes the mean of 20,000 simulated trajectories of 
400 observations each of a one-sided Cusum of independent normal 
increments with mean EZ[t] = (-0.1) and unit variance.  Started with any 
initial value, the mean of the Cusum should approach an asymptote as the 
number of observations increases;  when started at that asymptote, it 
should theoretically stay flat, unlike what we see here.

	  I would think this could be an artifact of the simulation 
methodology, but I've gotten essentially this image with several 
independently programmed simulations in S-Plus 6.1, with all six 
different random number generators in R 1.9.1 and 2.1.1 and with MS 
Excel.  For modest changes in EZ[t] < 0, I get a different asymptote but 
pretty much the same image.

#################################################
simCus5 <- function(mu=-0.1, Qp0=4.5, maxTime=400, nSims=20000){
   Qp.mean <- rep(NA, maxTime)
   Qp.t <- rep(Qp0, nSims)
   for(i in 1:maxTime){
     z.t <- (mu + rnorm(nSims))
     Qp.t <- pmax(0, Qp.t+z.t)
     Qp.mean[i] <- mean(Qp.t)
   }
   Qp.mean
}
set.seed(1)
plot(simCus5(Qp0=4.5))
#################################################

	  Thanks for your time in reading this.
	  Best Wishes,
	  Spencer Graves

Spencer Graves, PhD
Senior Development Engineer
PDF Solutions, Inc.
333 West San Carlos Street Suite 700
San Jose, CA 95110, USA

spencer.graves at pdf.com
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