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.. _continuous-studentized_range:
Studentized Range Distribution
==============================
This distribution has two shape parameters, :math:`k>1` and :math:`\nu>0`, and the support is :math:`x \geq 0`.
.. math::
:nowrap:
\begin{eqnarray*}
f(x; k, \nu) = \frac{k(k-1)\nu^{\nu/2}}{\Gamma(\nu/2)2^{\nu/2-1}}
\int_{0}^{\infty} \int_{-\infty}^{\infty} s^{\nu} e^{-\nu s^2/2} \phi(z) \phi(sx + z)
[\Phi(sx + z) - \Phi(z)]^{k-2} \,dz \,ds
\end{eqnarray*}
.. math::
:nowrap:
\begin{eqnarray*}
F(q; k, \nu) = \frac{k\nu^{\nu/2}}{\Gamma(\nu/2)2^{\nu/2-1}}
\int_{0}^{\infty} \int_{-\infty}^{\infty} s^{\nu-1} e^{-\nu s^2/2} \phi(z)
[\Phi(sq + z) - \Phi(z)]^{k-1} \,dz \,ds
\end{eqnarray*}
Note: :math:`\phi(z)` and :math:`\Phi(z)` represent the normal PDF and normal CDF, respectively.
When :math:`\nu` exceeds 100,000, the asymptotic approximation of :math:`F(x; k, \nu=\infty)` or :math:`f(x; k, \nu=\infty)` is used:
.. math::
:nowrap:
\begin{eqnarray*}
F(x; k, \nu=\infty) = k \int_{-\infty}^{\infty} \phi(z)
[\Phi(x + z) - \Phi(z)]^{k-1} \,dz
\end{eqnarray*}
.. math::
:nowrap:
\begin{eqnarray*}
f(x; k, \nu=\infty) = k(k-1) \int_{-\infty}^{\infty} \phi(z)\phi(x + z)
[\Phi(x + z) - \Phi(z)]^{k-2} \,dz
\end{eqnarray*}
Implementation: `scipy.stats.studentized_range`
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