On Thu, 21 Feb 2013 20:54:44 +0100, Cristiano <cristiapi@NSgmail.com> wrote:
>On 21/02/2013 6:46, Rich Ulrich wrote: >> On Wed, 20 Feb 2013 12:54:26 +0100, Cristiano <cristiapi@NSgmail.com> >> wrote: >> >>> Short question: does anybody know how to calculate the confidence >>> interval of the standard deviation for the uniform distribution? >>>
Oh, yeah, if I had read more carefully, I should have given you Ray's exact answer.... I figured you were wondering about the sampling distribution for small Ns, which might still be interesting (for instance, in design of experiments).
>>> >>> Long version. >>> >>> From a population of iid real numbers (double precision C++ type) I >>> randomly pick many numbers with replacement and I calculate the standard >>> deviation of those numbers. >>> >>> I repeat many times the above procedure to obtain many values of the >>> standard deviation. >>> >>> Then, I calculate the 10th percentile of the standard deviations. >>> >>> What's that percentile? I'm aware that it's not a confidence limit. >>> How do I calculate the CI via simulation? >>> >> >> I think I would call it a Monte Carlo estimate of the one-sided CI. > >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.
> >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 > >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.
> >> You could assume that the mean is known (since it is) and >> calculate the SDs while dividing by N instead of N-1, using >> that mean. > >I know both the mean and the standard deviation (of the population) >because I use real numbers in N(0,1) or U(0,1). > >Cristiano