Studentized Range (Percentile)

*The larger the sample size (r) and the smaller the degrees of freedom (v), the greater the error.

References)
Ferreira, D., et al. "Quantiles from the Maximum Studentized Range Distribution."
Biometric Brazilian Journal 25.1 (2007): 117-135.

\(\normalsize studentized\ range\ distribution\\ \hspace{5px} lower\ cumulative\ distribution\\ \hspace{30px} F(q,r,\nu)=P(Q\leq q) = {\large\int_{\small 0}^{\small \infty}} \left [ H \left ( q \sqrt{u} \right ) \right ]^{c} \hspace{2px} \large \frac{\nu^{\frac{\nu}{2}} e^{-\frac{u \nu}{2}} u^{\frac{\nu}{2}-1}} {2^{\frac{\nu}{2}} \Gamma \left( \frac{\nu}{2} \right)} du \\ \hspace{30px} H(q,r) = r {\large\int_{\small -\infty}^{\small \infty}} \theta (y) \left[ \Phi (y) - \Phi(y-q) \right]^{r-1} dy \\ \hspace{30px} \theta (y) = \large \frac{1}{\sqrt{2 \pi}} e^{\frac{-y^2}{2}} \\ \hspace{30px} \Phi(y) = {\large\int_{\small -\infty}^{\small y}} \theta (t) dt \\ \\ \normalsize q\ :\ vector\ of\ quantiles, \\ r\ :\ sample\ size\ for\ range,\\ \nu\ :\ degrees\ of\ freedom,\\ c\ :\ number\ of\ groups \)