스튜던트화 된 범위(백분위 점수)

※표본 크기 r가 크고, 자유도 v가 작을수록 오차가 커집니다.

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

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