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Let $f(m,n,w)$ be the probability density function of F variable with m numerator df and n denominator df, i.e.

$$f(m,n,w)=\frac{\Gamma\left(\frac{m+n}{2}\right)(m/n)^{m/2}}{\Gamma(m/2)\Gamma(n/2)}w^{(m/2)-1}\left(1+\frac{mw}{n}\right)^{-(m+n)/2}$$

I am interested in the infimum of $\int_1^\infty f(m,n,w) dw$ over all $m,n$. From my exploration, this seems to be $$1-erf(1/\sqrt{2})\approx 0.3173$$ where $erf(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2} dt$.

Can anyone prove this or point me to a reference?

  • 0
    What is the significance of this result?2011-03-11
  • 0
    This is related to testing hypothesis about equality of variances.2011-03-11
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    Do you require $m$ and $n$ to be positive integers?2011-03-11
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    Yes. They are positive integers. It appears that the inf (min) attains at m=1, n=infinity, i.e. when it becomes $\chi^2$ with one degree of freedom.2011-03-11
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    I think your conjecture is correct, but unfortunately I don't see how to prove it, either.2011-03-11

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