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Re: Simulation for the standard deviation
Posted:
Feb 27, 2013 3:09 AM
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On Feb 26, 5:36 pm, Cristiano <cristi...@NSgmail.com> wrote:
> On 26/02/2013 1:56, David Jones wrote:
>
>> The standard work, Kendall & Stuart's "Theoretical Statistics" will
>> provide a formula for variance of the sample variance. This could be
>> used to provide a sampling interval for the sample variance, and hence
>> for the sample standard deviation. Indeed, a formula for the variance of
>> the sample variance exists on the Wikipedia page for "Variance".
>>
>> In the present context, an alternative is just to do multiple
>> simulations of the sample standard deviation and use this to get the
>> sampling distribution empirically. Of course this would assume that the
>> random number generator has adequate properties.
>
> In my simulations I use the (very good and very fast) dSFMT generator
> and I get, for example, with n= 2, E[s^2]= .0833584 (good) and Var[s^2]=
> .00972501.
>
> Using the formula given here:
> http://en.wikipedia.org/wiki/Variance#Distribution_of_the_sample_variance
> I get: Var[s^2]= .00369776.
>
> Using the formula given here:
> http://mathworld.wolfram.com/SampleVarianceDistribution.html
> I get: Var[s^2]= .000924963.
>
> Where is the error?
>
> Cristiano
The variance of the unbiased sample variance in samples of size n
from a Uniform(0,1) distribution is (2n+3)/(360n(n-1)). For n = 2
this reduces to 7/720 = .0097222... .
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