On Tue, 27 Jun 2006, Christian Hennig wrote: > Dear list, > > I did simulations in which I generated 10000 > independent Bernoulli(0.5)-sequences of length 100. I estimated > p for each sequence and I also estimated the conditional probability that > a one is followed by another one (which should be p as well). > However, the second probability is significantly smaller than 0.5 (namely > about 0.494, see below) and of course smaller than the direct estimates of > p as well, indicating negative correlation between the random numbers. > > See below the code and the results. > Did I do something wrong or are the numbers in > fact negatively correlated? (A type I error is quite unlikely with a > p-value below 2.2e-16.)
I think you did something wrong, and that there is a problem with overlapping blocks of two. If you do x<-matrix(rbinom(1e6,1,p),ncol=2) tt<-table(x[,1],x[,2]) you get much better looking results. In this case you have 500,000 independent pairs of numbers that can be 01, 10, 11, 00. A test for independence seems fine. > tt 0 1 0 125246 124814 1 125140 124800 > fisher.test(tt) Fisher's Exact Test for Count Data data: tt p-value = 0.8987 alternative hypothesis: true odds ratio is not equal to 1 95 percent confidence interval: 0.9896745 1.0119211 sample estimates: odds ratio 1.000735 In your case the deficit in est11 is suspiciously close to 0.5/n. Changing n to 1000 and using the same seed I get > mean(est11)-0.5 [1] -0.0005534743 10 times smaller, and still close to 0.5/n. Now, consider what happens in a case where we can see all the possibilities, n=3 x 1/0 1/1 000 - - 001 - - 010 1 0 011 0 1 100 1 0 101 1 0 110 1 1 111 2 0 So that if each of these three triplets has the same probability your est11 would be 2/8 rather than 4/8, and est11 is not an unbiased estimate of the long-run conditional probability. The bias is of order 1/n, so you need n to be of larger order than sqrt(simruns). -thomas ______________________________________________ R-help@stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html