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Topic: Chebyshev Inequality for Sample Variance
Replies: 5   Last Post: Oct 5, 2010 9:23 AM

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C Hanck

Posts: 48
Registered: 7/29/07
Re: Chebyshev Inequality for Sample Variance
Posted: Oct 2, 2010 4:05 AM
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On Oct 2, 2:09 am, Cagdas Ozgenc <cagdas.ozg...@gmail.com> wrote:
> On 1 Ekim, 20:15, Ludovicus <luir...@yahoo.com> wrote:
>

> > On Sep 24, 9:42 am, Cagdas Ozgenc <cagdas.ozg...@gmail.com> wrote:
>
> > > How do you adjust Chebyshev Inequality for Sample Variance when
> > > Population Variance is not known?

>
> > That's impossible because Chebyshev Inequality is an arithmetic
> > theorem applied to Probability based in the sample variance.
> > Chebyshev Theorem:
> > "Given any set of of numbers with Standard deviation s, the fraction
> > that deviates more than k.s from the mean is always less than 1/k^2
> > Ludovicus

>
> Of course possible. I found a paper regarding this matter,
> unfortunately I don't have access to it.
>
> http://www.jstor.org/pss/2683249


If you know that it is a random sample and the sufficiently high
moments exist, then

P(|S^2-\sigma^2|>eps)<Var(S^2)/eps^2->0

as n->oo because the variance of S^2 will then tend to zero.

Also the claim about Chebychev's inequality being about the sample
mean is wrong, this is just the most prominent application. In
general, it says that the probability that some r.v. is more than an
epsilon away from another in absolute value is bounded from above by
the expected value of the squared deviation divided by epsilon^2.



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