On Fri, 22 Feb 2013 15:12:03 +0100, Cristiano <cristiapi@NSgmail.com> wrote:
>On 22/02/2013 0:52, Rich Ulrich wrote: >> On Thu, 21 Feb 2013 20:54:44 +0100, Cristiano <cristiapi@NSgmail.com> >> wrote: >> >>> To check my simulation, I calculated the CI for the normal distribution >>> as explained here: >>> http://www.itl.nist.gov/div898/handbook/prc/section2/prc231.htm >> >> >> CAREFUL. You stated a problem about the uniform. You are >> pointing now to an answer about the normal? Not relevant. > >I know how to exactly calculate the confidence interval for the normal >distribution, so I use the exact confidence interval to check my >simulation. I'd say that it is relevant! :-)
Okay -- So you needed to make it evident to me that you were testing the simulation, and not comparing outcomes... since I (still) don't follow your comparison.
> >>> For example, for N(0,1), when N= 60 and alpha= 0.01, I get: >>> 0.806465 <= sigma <= 1.30263 (two-sided) >>> and >>> 0.822722 <= sigma, 0 <= sigma <= 1.26795
Puzzled out, maybe. The first line has the two 0.5% one-sided limits -- which is how you create the two-sided CI -- and the second line has the two 1% one-sided limits. I'm not bothering to compute it to find out.
>>> >>> while the simulation converges to 0.7885 <= sigma. >>> >>>> Assuming that you are describing a recommended procedure for >>>> boot-strapping, that seems like the way to simulate that CI. >>> >>> I found this link: >>> http://en.wikipedia.org/wiki/Bootstrapping_%28statistics%29#Deriving_confidence_intervals_from_the_bootstrap_distribution >>> and I think that I'm using the "Percentile Bootstrap". >>> >>> There is an interesting phrase: "This method can be applied to any >>> statistic. It will work well in cases where the bootstrap distribution >>> is symmetrical and centered on the observed statistic [...]". >>> >>> The reason of my wrong result could be that the bootstrap distribution >>> of the standard deviation is not symmetrical, I guess. >> >> When you are looking at one tail only, the matter of symmetry does >> not arise. You stated your tentative solution as just one percentile. >> The difficulty of asymmetry does not arise for one limit alone. > >Then there should be something wrong in my simulation, because I get a >totally wrong result for the one-sided CI of the SD for the normal >distribution: 0.7885 instead of 0.8227. >
Those numbers that you list above are limits around sigma/s rather than s/sigma -- You took the formula at .nist and divided through by s. Personally, I was surprised that the formula seemed to be upside down. I was expecting s/sigma. Now I wonder which should be used for what.
However, when I take the reciprocal of 1.26795, it comes out as 0.7887 -- which makes me think that the only error you see is computation error on someone's CDF generator. Cumulative round-off in the simulation seems possible, but less likely if you did enough steps to converge.