I want to analyze the $1-\alpha$-quantile, $\alpha\in(0,1)$, of a $F_{n, m}$ distribution, keeping n fixed while increasing m. It seems that the quantile decreases monotonically, but I would like to be sure about this and furthermore know if it converges to a specific value.
I am aware that the cumulative distribution function is given by an incomplete beta function. However, inverting it to get the quantile function does not seem trivial. Are there existing results on this?
$\newcommand\al\alpha$If the $(1-\al)$-quantiles of $F_{n,m}$ were decreasing monotonically in $m$ for each $\al\in(0,1)$, the the corresponding cdf's -- say $G_{m,n}$ -- would be increasing pointwise on $(0,\infty)$ monotonically in $m$. However, this is not so, as seen from the graphs below of $G_{3,1}-G_{3,1}$ (red), $G_{3,2}-G_{3,1}$ (orange), $G_{3,3}-G_{3,1}$ (green), $G_{3,4}-G_{3,1}$ (blue), $G_{3,5}-G_{3,1}$ (magenta):
However, recall that $F_{n,m}$ is the distribution of $$R_{n,m}:=\frac{X_n/n}{Y_m/m},$$ where $X_n$ and $Y_m$ are independent chi-squared random variables with $n$ and $m$ degrees of freedom, respectively.
By the law of large numbers, $Y_m/m\to1$ in probability as $m\to\infty$. So, as $m\to\infty$, $F_{n,m}$ converges weakly to the distribution of $X_n/n$, which is the gamma distribution with parameters $n/2,2/n$.
