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Suppose $X\sim F(a,b)$. Is there any sharp upper bound of the following probability with large $x$? $$\mathbb{P}(X\geq x)$$ what is the order of the above probability as $x\to+\infty$?

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    $\begingroup$ What's $F(a,b)$? $\endgroup$ Oct 11, 2018 at 23:58
  • $\begingroup$ Do you have a response to the answer below? $\endgroup$ Feb 17, 2022 at 14:06

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The pdf of the $F$ distribution $F_{a,b}$ with $a,b$ degrees of freedom is given by \begin{equation} f(x)= c\, x^{a/2-1} (a x+b)^{-a/2-b/2} \end{equation} for $x>0$, where $c:=a^{a/2} b^{b/2}/B\left(\frac{a}{2},\frac{b}{2}\right)$. Replacing here $a x+b$ by $ax$, we get the upper bound \begin{equation} g(x):=ca^{-a/2-b/2}\, x^{-1-b/2} \end{equation} on $f(x)$, which is also asymptotic to $f(x)$ (as $x\to\infty$). So, \begin{equation} G(x):=\int_x^\infty g(u)\,du=2ca^{-a/2-b/2}\, x^{-b/2}/b \end{equation} is an upper bound on the tail probability $P(X\ge x)$ for $X\sim F_{a,b}$ and $x>0$. Also, by l'Hospital's rule, this bound is asymptocally exact: \begin{equation} P(X\ge x)\sim G(x). \end{equation}

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