Calculate the critical value of Tukey q For the life of me, I can not find an equation for the PDF or CDF of the studentized range distribution. Is there a closed-form PDF that I can numerically integrate? Is there a infinite series approximation that I can use?
 A: The Studentized Range Distribution is a function of q, k, and df, where k is the number of groups of means, and df is the degrees of freedom. If $\phi(z)$ is the standard normal PDF, and $\Phi(z)$ is the standard normal CDF:
$$RangeCDF(q,k,\infty) = k \int_{-\infty}^\infty\phi(z)[\Phi(z+q)-\Phi(z)]^{k-1}dz$$
This seems to agree with the tables of critical q values when $df=\infty$. However, if I replace the standard normal with Student's T, the calculated value does not match the table, except when $df \to \infty$.
$$Studentized RangeCDF(q,k,df) = k \int_{-\infty}^\infty t(t,df)[T(t+q,df)-T(t,df)]^{k-1}dt$$
Edit:
I now have correct values when k=2, or df=$\infty$:
$$2* \int_{-\infty}^\infty\phi(z)[\Phi(z+q)-\Phi(z)]^{2-1}dz = 2 \int_{-\infty}^\infty\phi(z)*\Phi(z+q)-\phi(z)\Phi(z)dz$$
$$=2*[\int_{-\infty}^\infty\phi(z)\Phi(z+q)dz-\int_{-\infty}^\infty\phi(z)\Phi(z)dz]$$
if $u=\Phi(z); du=\phi(z)dz$
$$=2*[\int_{-\infty}^\infty\phi(z)\Phi(z+q)dz-\int_{-\infty}^\infty udu]$$
$$=2*[\int_{-\infty}^\infty\phi(z)*\Phi(z+q)dz-\frac{1}{2}(\Phi^2(\infty)-\Phi^2(-\infty)]$$
$$=2*\left[\int_{-\infty}^\infty\phi(z)*\Phi(z+q)dz-\frac{1}{2}\right]$$
$$=2*\left[\Phi\left(\frac{q}{\sqrt2}\right)-\frac{1}{2}\right]$$
Using Student's T distribution with the specified degrees of freedom yields the same results as the reference table. However, my numerical integration of the second (StudentizedRange) equation does not match. This must mean that the process used to integrate $\phi(z)\Phi(z+q)dz$ does not work for $t$ and $T$
