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Distributions related to multiple comparisons of means
==========================================================
.. _rst_nmax_rho_pdf:
.. _rst_nmax_rho_cdf:
Normal maximum distribution, `\rho_{ij, i \ne j} = \rho`: pdf, cdf, qtf, boost class
-------------------------------------------------------------------------------------------
.. method:: math53lib.NmaxRhoPdf(x, rho, k)
.. method:: math53lib.NmaxRhoCdf(x, rho, k)
.. method:: math53lib.NmaxRhoQtf(q, rho, k)
.. method:: math53lib.NmaxRhoDist(rho, k)
Returns the pdf, cdf, qtf or boost class of a random variable `X`, following the distribution of the maximum of `k \ge 2` correlated standard normal variates, with common correlation `-1/(k-1) \le \rho \le 1` and the support interval `(-\infty, +\infty)`, and `0 \le q \le 1`.
See also :cite:t:`Dunnett1955`, :cite:t:`Bechhofer1988`, :cite:t:`Grubbs1969`, :cite:t:`Bechhofer1988`, :cite:t:`Stoline1979`, and :cite:t:`Hahn1971`.
.. math:: \text{pdf}(x) = f_{\text{nmaxrho}}(x, \rho, k) = \frac{k}{b} \int_{-\infty}^\infty \left( \Phi(z_1) \right)^{k-1} \phi(z_1) \: \phi(y) \: dy,
.. math:: \text{cdf}(x)= F_{\text{nmaxrho}}(x, \rho, k) = \int_{-\infty}^\infty \left( \Phi(z_1) \right)^{k} \phi(y) dy,
.. math:: \text{sf}(x) = 1 - \int_{-\infty}^\infty \left( \Phi(z_1) \right)^{k} \phi(y) dy,
where `\displaystyle z_1 = \frac{x+a}{b}`, `a = y \sqrt{\rho}`, and `b = \sqrt{1-\rho}`. Note that while `z_1` is a complex number for `\rho < 0`, the integral always evaluates to a real number. In the context of the Dunnett t-test with reference sample size `n_0` and common comparator sample size `n_i`, `\displaystyle \rho=\frac{1}{1+n_0/n_i}`. In the context of maximum deviation from the common mean (equal sample sizes), `\displaystyle \rho=-\frac{1}{k-1}`.
There is no known explicit expression for `\text{qtf}(q)` or `\text{isf}(q)`: These functions are computed with Newton iterations where the starting values are from the corresponding Boost functions (in double precision).
An example in Python
.. code-block:: pycon
>>> from mpfebnet import XReal
>>> XReal.NmaxRhoPdf(x, rho, k)
XReal('5.2359877559829887307E-1')
An example in Visual Basic
.. code-block:: pycon
>>> from mpfebnet import XReal
>>> XReal.NmaxRhoPdf(x, rho, k)
XReal('5.2359877559829887307E-1')
>>> from mpfebnet import XReal
>>> XReal.NmaxRhoCdf(x, rho, k)
XReal('5.2359877559829887307E-1')
>>> XReal.NmaxRhoCdf(x, rho, k)
XReal('5.3518479027559984754E-1')
>>> from mpfebnet import XReal
>>> XReal.NmaxRhoQtf(q, rho, k)
XReal('5.2359877559829887307E-1')
>>> XReal.NmaxRhoQtf(q, rho, k)
XReal('5.3518479027559984754E-1')
|newpage|
.. _rst_nmm_rho_pdf:
.. _rst_nmm_rho_cdf:
Normal maximum modulus distribution, `\rho_{ij, i \ne j} = \rho`: pdf, cdf, qtf, boost class
--------------------------------------------------------------------------------------------------
.. method:: math53lib.NmmRhoPdf(x, rho, k)
.. method:: math53lib.NmmRhoCdf(x, rho, k)
.. method:: math53lib.NmmRhoQtf(q, rho, k)
.. method:: math53lib.NmmRhoDist(rho, k)
Returns the pdf, cdf, qtf or boost class of a random variable `X`, following the distribution of the maximum of the absolute value of `k \ge 2` correlated standard normal variates, with common correlation `-1/(k-1) \le \rho \le 1` and the support interval `(0, +\infty)`, and `0 \le q \le 1`.
See also :cite:t:`Dunnett1955`, :cite:t:`Bechhofer1988`, :cite:t:`Grubbs1969`, :cite:t:`Bechhofer1988`, :cite:t:`Stoline1979`, and :cite:t:`Hahn1971`.
.. math:: \text{pdf}(x) = \frac{k}{b} \int_{-\infty}^\infty \left( \Phi(z_1)- \Phi(z_2) \right)^{k-1} (\phi(z_1)+\phi(z_1)) \: \phi(y) \: dy,
.. math:: \text{cdf}(x)= F_{\text{nmaxrho}}(x, \rho, k) = \int_{-\infty}^\infty \left( \Phi(z_1) - \Phi(z_2) \right)^{k} \phi(y) dy,
.. math:: \text{sf}(x) = 1 - \int_{-\infty}^\infty \left( \Phi(z_1) - \Phi(z_2) \right)^{k} \phi(y) dy,
where `\displaystyle z_1 = \frac{x+a}{b}`, `\displaystyle z_2 = \frac{-x+a}{b}`, `a = y \sqrt{\rho}`, and `b = \sqrt{1-\rho}`. Note that while `z_1` and `z_2` are complex numbers for `\rho < 0`, the integral always evaluates to a real number. In the context of the Dunnett t-test with reference sample size `n_0` and common comparator sample size `n_i`, `\displaystyle \rho=\frac{1}{1+n_0/n_i}`. In the context of maximum deviation from the common mean (equal sample sizes), `\displaystyle \rho=-\frac{1}{k-1}`.
There is no known explicit expression for `\text{qtf}(q)` or `\text{isf}(q)`: These functions are computed with Newton iterations where the starting values are from the corresponding Boost functions (in double precision).
An example in Python
.. code-block:: pycon
>>> from mpfebnet import XReal
>>> XReal.NmmRhoPdf(x, rho, k)
XReal('5.2359877559829887307E-1')
An example in Visual Basic
.. code-block:: pycon
>>> from mpfebnet import Gpr
>>> Gpr.NmmRhoPdf(x, rho, k)
Gpr('5.2359877559829887307E-1')
>>> from mpfebnet import XReal
>>> XReal.NmmRhoCdf(x, rho, k)
XReal('5.2359877559829887307E-1')
>>> XReal.NmmRhoCdf(x, rho, k)
XReal('5.3518479027559984754E-1')
>>> from mpfebnet import XReal
>>> XReal.NmmRhoQtf(q, rho, k)
XReal('5.2359877559829887307E-1')
>>> XReal.NmmRhoQtf(q, rho, k)
XReal('5.3518479027559984754E-1')
|newpage|
.. _rst_nmax_rhoij_pdf:
.. _rst_nmax_rhoij_cdf:
Normal maximum distribution, `\rho_{ij, i \ne j} = \lambda_i \lambda_j`: pdf, cdf, qtf, boost class
------------------------------------------------------------------------------------------------------
.. method:: math53lib.NmaxRhoijPdf(x, lambda_i, k)
.. method:: math53lib.NmaxRhoijCdf(x, lambda_i, k)
.. method:: math53lib.NmaxRhoijQtf(q, lambda_i, k)
.. method:: math53lib.NmaxRhoijDist(lambda_i, k)
Returns the pdf, cdf, qtf or boost class of a random variable `X`, following the distribution of the maximum deviate of `k \ge 2` correlated standard normal variates, with common correlation `0 \le \rho \le 1` and the support interval `(-\infty, +\infty)`, and `0 \le q \le 1`.
See also :cite:t:`Dunnett1955`, :cite:t:`Bechhofer1988`, :cite:t:`Grubbs1969`, :cite:t:`Bechhofer1988`, :cite:t:`Stoline1979`, and :cite:t:`Hahn1971`.
.. math:: \text{pdf}(x) = f_{\text{nmaxrhoij}}(x, \rho, k) = \frac{k}{b} \int_{-\infty}^\infty \left( \Phi(z_1) \right)^{k-1} \phi(z_1) \: \phi(y) \: dy, \quad \text{where}
.. math:: z_1 = \frac{x+a}{b}, \quad a = y \sqrt{|\rho|}, \quad b = \sqrt{1-\rho}, \rho = -\frac{1}{k-1}.
.. math:: \text{cdf}(x)= F_{\text{nmaxrhoij}}(x, \rho, k) = \int_{-\infty}^\infty \left[\Phi \left(\frac{x + \sqrt{\vert \rho \vert} y} {\sqrt{1-\rho}} \right) \right]^n \phi(y) dy , \rho = -\frac{1}{k-1}.
.. math:: \text{sf}(x) = 1 - \int_{-\infty}^\infty \left[\Phi \left(\frac{x + \sqrt{\vert \rho \vert} y} {\sqrt{1-\rho}} \right) \right]^n \phi(y) dy , \rho = -\frac{1}{k-1}.
For `\lambda_i` = `\sqrt{\rho} \geq 0` for all `i`, this reduces to the equicorrelated case.
With `\rho_{ij}=\lambda_i \lambda_j`, we have for a one-sided test:
.. math:: F_n(h;\rho_{ij}) = \int_{-\infty}^\infty \prod_{i=1}^n \left[\Phi \left(\frac{(a_i-\mu_i)/\sigma_i + \lambda_i z} {\sqrt{1-\lambda_i^2}} \right) \right] \phi(y) dz
There is no known explicit expression for `\text{qtf}(q)` or `\text{isf}(q)`: These functions are computed with Newton iterations where the starting values are from the corresponding Boost functions (in double precision).
An example in Python
.. code-block:: pycon
>>> from mpfebnet import XReal
>>> XReal.NmaxRhoijPdf(q, rho, k)
XReal('5.2359877559829887307E-1')
>>> XReal.NmaxRhoijPdf(q, rho, k)
XReal('5.3518479027559984754E-1')
An example in Visual Basic
.. code-block:: pycon
>>> from mpfebnet import Gpr
>>> Gpr.NmaxRhoijPdf(q, rho, k)
Gpr('5.2359877559829887307E-1')
>>> Gpr.NmaxRhoijPdf(q, rho, k)
Gpr('5.3518479027559984754E-1')
>>> from mpfebnet import XReal
>>> XReal.NmaxRhoijCdf(q, rho, k)
XReal('5.2359877559829887307E-1')
>>> XReal.NmaxRhoijCdf(q, rho, k)
XReal('5.3518479027559984754E-1')
>>> from mpfebnet import XReal
>>> XReal.NmaxRhoijQtf(q, rho, k)
XReal('5.2359877559829887307E-1')
>>> XReal.NmaxRhoijQtf(q, rho, k)
XReal('5.3518479027559984754E-1')
|newpage|
.. _rst_nmm_rhoij_pdf:
.. _rst_nmm_rhoij_cdf:
Normal maximum modulus distribution, `\rho_{ij, i \ne j} = \lambda_i \lambda_j`: pdf, cdf, qtf, boost class
---------------------------------------------------------------------------------------------------------------
.. method:: math53lib.NmmRhoijPdf(x, rho, k)
.. method:: math53lib.NmmRhoijCdf(x, rho, k, cdf=True)
.. method:: math53lib.NmmRhoijQtf(q, rho, k, qtf=True)
.. method:: math53lib.NmmRhoijDist(q, rho, k, qtf=True)
Returns the pdf, cdf, qtf or boost class of a random variable `X`, following the distribution of the maximum of `k \ge 2` correlated standard normal variates, with correlation `\rho_{ij, i \ne j} = \lambda_i \lambda_j` and the support interval `(-\infty, +\infty)`, and `0 \le q \le 1`.
See also :cite:t:`Dunnett1955`, :cite:t:`Bechhofer1988`, :cite:t:`Grubbs1969`, :cite:t:`Bechhofer1988`, :cite:t:`Stoline1979`, and :cite:t:`Hahn1971`.
.. math:: \text{pdf}(x) = \frac{k}{b} \int_{-\infty}^\infty \left( \Phi(z_1)- \Phi(z_2) \right)^{k-1} (\phi(z_1)+\phi(z_1)) \: \phi(y) \: dy, \quad \text{where }
.. math:: z_1 = \frac{x+a}{b}, z_2 = \frac{-x+a}{b}, \quad a = y \sqrt{|\rho|}, \quad b = \sqrt{1-\rho}, \rho = -\frac{1}{k-1}.
.. math:: \text{cdf}(x) = \int_{-\infty}^\infty \left[\Phi \left(\frac{x + \sqrt{\vert \rho \vert} y} {\sqrt{1-\rho}} \right) - \Phi \left(\frac{-x + \sqrt{\vert \rho \vert} y} {\sqrt{1-\rho}} \right) \right]^k \phi(y) dy , \rho = -\frac{1}{k-1}.
.. math:: \text{sf}(x) = 1 - \int_{-\infty}^\infty \left[\Phi \left(\frac{x + \sqrt{\vert \rho \vert} y} {\sqrt{1-\rho}} \right) - \Phi \left(\frac{-x + \sqrt{\vert \rho \vert} y} {\sqrt{1-\rho}} \right) \right]^k \phi(y) dy , \rho = -\frac{1}{k-1}.
For `\lambda_i` = `\sqrt{\rho} \geq 0` for all `i`, this reduces to the equicorrelated case.
With `\rho_{ij}=\lambda_i \lambda_j`, we have for a two-sided test:
.. math:: F_n(h;\rho_{ij}) = \int_{-\infty}^\infty \prod_{i=1}^n \left[\Phi \left(\frac{(a_i-\mu_i)/\sigma_i + \lambda_i z} {\sqrt{1-\lambda_i^2}} \right) - \Phi \left(\frac{(b_i-\mu_i)/\sigma_i + \lambda_i z} {\sqrt{1-\lambda_i^2}} \right) \right] \phi(y) dz
There is no known explicit expression for `\text{qtf}(q)` or `\text{isf}(q)`: These functions are computed with Newton iterations where the starting values are from the corresponding Boost functions (in double precision).
An example in Python
.. code-block:: pycon
>>> from mpfebnet import XReal
>>> XReal.NmmRhoijPdf(x, rho, k)
XReal('5.2359877559829887307E-1')
>>> XReal.NmmRhoijPdf(x, rho, k)
XReal('5.3518479027559984754E-1')
An example in Visual Basic
.. code-block:: pycon
>>> from mpfebnet import Gpr
>>> Gpr.NmmRhoijPdf(x, rho, k)
Gpr('5.2359877559829887307E-1')
>>> Gpr.NmmRhoijPdf(x, rho, k)
Gpr('5.3518479027559984754E-1')
>>> from mpfebnet import XReal
>>> XReal.NmmRhoijCdf(x, rho, k)
XReal('5.2359877559829887307E-1')
>>> XReal.NmmRhoijCdf(x, rho, k)
XReal('5.3518479027559984754E-1')
>>> from mpfebnet import XReal
>>> XReal.NmmRhoijQtf(q, rho, k)
XReal('5.2359877559829887307E-1')
>>> XReal.NmmRhoijQtf(q, rho, k)
XReal('5.3518479027559984754E-1')
|newpage|
.. _rst_mpm_nrange_pdf:
.. _rst_mpm_nrange_cdf:
Normal range distribution: pdf, cdf, qtf, boost class
-------------------------------------------------------------------------------
.. method:: math53lib.NrangePdf(x, k)
.. method:: math53lib.NrangeCdf(n, x)
.. method:: math53lib.NrangeQtf(q, k)
.. method:: math53lib.NrangeDist(k)
Returns the pdf, cdf, qtf or boost class of a random variable `X`, following a normal range distribution, with `k \ge 2` groups, and the support interval `(0, +\infty)`, and `0 \le q \le 1`.
See also Wikipedia :cite:p:`WikipediaDis60`, :cite:t:`Harter1960`, :cite:t:`RDis60`.
The density in the central case is given by
.. math:: \text{pdf}(x) = k(k-1) \int_{-\infty}^\infty \left( \Phi(y) - \Phi(y-x) \right)^{k-2} \phi(y) \phi(y-x) dy
We have
.. math:: \text{cdf}(x) = k \int_{-\infty}^\infty \phi(y) \left( \Phi(y) - \Phi(y-x) \right)^{k-1} dy
.. math:: \text{sf}(x) = 1 - k \int_{-\infty}^\infty \phi(y) \left( \Phi(y) - \Phi(y-x) \right)^{k-1} dy
In the noncentral case, this becomes
.. math:: \text{cdf}(x) = F_sr \left(m,n,x\right) = \sum_{i=1}^k \int_{-\infty}^\infty \phi(y_i - \mu_i) \left( \prod_{i=1, j \neq i}^k (\Phi(y_i - \mu_i) - \Phi(y_i - \mu_i -x) \right) dy_i.
There is no known explicit expression for `\text{qtf}(q)` or `\text{isf}(q)`: These functions are computed with Newton iterations where the starting values are from the corresponding Boost functions (in double precision).
An example in Python
.. code-block:: pycon
>>> from mpfebnet import XReal
>>> XReal.NrangePdf(x, k)
XReal('5.2359877559829887307E-1')
>>> XReal.NrangePdfxq, k)
XReal('5.3518479027559984754E-1')
An example in Visual Basic
.. code-block:: pycon
>>> from mpfebnet import Gpr
>>> Gpr.NrangePdf(x, k)
Gpr('5.2359877559829887307E-1')
>>> Gpr.NrangePdfxq, k)
Gpr('5.3518479027559984754E-1')
>>> from mpfebnet import XReal
>>> XReal.NrangeCdf(x, k)
XReal('5.2359877559829887307E-1')
>>> XReal.NrangePdf(x, k)
XReal('5.3518479027559984754E-1')
>>> from mpfebnet import XReal
>>> XReal.NrangeQtf(q, k)
XReal('5.2359877559829887307E-1')
>>> XReal.NrangeQtf(q, k)
XReal('5.3518479027559984754E-1')
|newpage|
Studentized maximum distribution: pdf, cdf, qtf, boost class
-------------------------------------------------------------------------------
.. method:: math53lib.SmaxPdf(x, k, nu)
.. method:: math53lib.SmaxCdf(x, k, nu)
.. method:: math53lib.SmaxQtf(q, k, nu)
.. method:: math53lib.SmaxDist(k, nu)
Returns the pdf, cdf, qtf or boost class of a random variable `X`, following a studentized maximum distribution with `k \ge 1` groups, `\nu` error degrees of freedom, and the support interval `(-\infty, +\infty)`, and `0 \le q \le 1`.
See also :cite:t:`Stoline1979`, :cite:t:`Hochberg1987`, :cite:t:`Narula1978`.
.. math:: \text{pdf}(x) = \int_{0}^\infty f_{\text{nnax}}(sx, k) \cdot s \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds
where `f_{\text{nmax}}(\cdot, k)` is the pdf of the normal maximum with `k` groups, and `f_{\chi}(\cdot, \nu)` is the pdf of the `\chi`-distribution with `\nu` degrees of freedom.
.. math:: \text{cdf}(x) = \int_{0}^\infty F_{\text{nmax}}(sx, k) \cdot \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds
where `F_{\text{nnax}}(\cdot, k)` is the cdf of the normal maximum with `k` groups, and `f_{\chi}(\cdot, \nu)` is the pdf of the `\chi`-distribution with `\nu` degrees of freedom.
.. math:: \text{sf}(x) = 1 - \text{cdf}(x)
There is no known explicit expression for `\text{qtf}(q)` or `\text{isf}(q)`: These functions are computed with Newton iterations where the starting values are from the corresponding Boost functions (in double precision).
An example in Python
.. code-block:: pycon
>>> from mpfebnet import XReal
>>> XReal.SmaxPdf(x, k)
XReal('5.2359877559829887307E-1')
>>> XReal.SmaxPdf(x, k)
XReal('5.3518479027559984754E-1')
An example in Visual Basic
.. code-block:: pycon
>>> from mpfebnet import Gpr
>>> Gpr.SmaxPdf(x, k)
Gpr('5.2359877559829887307E-1')
>>> Gpr.SmaxPdf(x, k)
Gpr('5.3518479027559984754E-1')
>>> from mpfebnet import XReal
>>> XReal.SmaxCdf(x, k)
XReal('5.2359877559829887307E-1')
>>> XReal.SmaxCdf(x, k)
XReal('5.3518479027559984754E-1')
>>> from mpfebnet import XReal
>>> XReal.SmaxQtf(q, k)
XReal('5.2359877559829887307E-1')
>>> XReal.SmaxQtf(q, k)
XReal('5.3518479027559984754E-1')
|newpage|
Studentized maximum modulus distribution: pdf, cdf, qtf, boost class
-------------------------------------------------------------------------------
.. method:: math53lib.SmmPdf(x, k, nu)
.. method:: math53lib.SmmCdf(x, k, nu)
.. method:: math53lib.SmmQtf(q, k, nu)
.. method:: math53lib.SmmDist(k, nu)
Returns the pdf, cdf, qtf or boost class of a random variable `X`, of a random variable `X`, following a studentized maximum modulus distribution with `k \ge 1` groups, `\nu` error degrees of freedom, and the support interval `(0, +\infty)`, and `0 \le q \le 1`.
See also :cite:t:`Stoline1979`, :cite:t:`Hochberg1987`, :cite:t:`Narula1978`.
.. math:: \text{pdf}(x) = \int_{0}^\infty f_{\text{nmm}}(sx, k) \cdot s \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds
where `f_{\text{nmm}}(\cdot, k)` is the pdf of the normal maximum modulus with `k` groups, and `f_{\chi}(\cdot, \nu)` is the pdf of the `\chi`-distribution with `\nu` degrees of freedom.
.. math:: \text{cdf}(x) = \int_{0}^\infty F_{\text{nmm}}(sx, k) \cdot \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds
where `F_{\text{nmm}}(\cdot, k)` is the cdf of the normal maximum modulus with `k` groups, and `f_{\chi}(\cdot, \nu)` is the pdf of the `\chi`-distribution with `\nu` degrees of freedom.
.. math:: \text{sf}(x) = 1 - \text{cdf}(x)
There is no known explicit expression for `\text{qtf}(q)` or `\text{isf}(q)`: These functions are computed with Newton iterations where the starting values are from the corresponding Boost functions (in double precision).
An example in Python
.. code-block:: pycon
>>> from mpfebnet import XReal
>>> XReal.SmmPdf(x, k, nu)
XReal('5.2359877559829887307E-1')
>>> XReal.SmmPdf(x, k, nu)
XReal('5.3518479027559984754E-1')
An example in Visual Basic
.. code-block:: pycon
>>> from mpfebnet import XReal
>>> XReal.SmmPdf(x, k, nu)
XReal('5.2359877559829887307E-1')
>>> XReal.SmmPdf(x, k, nu)
XReal('5.3518479027559984754E-1')
>>> from mpfebnet import XReal
>>> XReal.SmmCdf(x, k, nu)
XReal('5.2359877559829887307E-1')
>>> XReal.SmmCdf(x, k, nu)
XReal('5.3518479027559984754E-1')
>>> from mpfebnet import XReal
>>> XReal.SmmQtf(q, k, nu)
XReal('5.2359877559829887307E-1')
>>> XReal.SmmQtf(q, k, nu)
XReal('5.3518479027559984754E-1')
|newpage|
Dunnett `t`-distribution, 1-sided: pdf, cdf, qtf, boost class
-------------------------------------------------------------------------------
.. method:: math53lib.Dunnett1Pdf(x, rho, k, nu)
.. method:: math53lib.Dunnett1Cdf(x, rho, k, nu)
.. method:: math53lib.Dunnett1Qtf(q, rho, k, nu)
.. method:: math53lib.Dunnett1Dist(rho, k, nu)
Returns the pdf, cdf, qtf or boost class of a random variable `X`, of a random variable `X`, following a 1-sided Dunnett `t`-distribution with common correlation `\rho`, `k \ge 2` groups (including control group), error degrees of freedom `\nu`, and the support interval `(-\infty, +\infty)`, and `0 \le q \le 1`.
See also :cite:t:`Dunnett1955`, :cite:t:`Bechhofer1988`.
.. math:: \text{pdf}(x) = \int_{0}^\infty f_{\text{nmaxrho}}(sx, \rho, k) \cdot s \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds
where `f_{\text{nmaxrho}}(\cdot, \rho, k)` is the pdf of the normal maximum (equicorrelated case) with common correlation `\rho` and `k` groups (see :ref:`nmax_rho_pdf() `), and `f_{\chi}(\cdot, \nu)` is the pdf of the `\chi`-distribution with `\nu` degrees of freedom.
.. math:: \text{cdf}(x) = \int_{0}^\infty F_{\text{nmaxrho}}(sx, \rho, k) \cdot \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds
where `F_{\text{nmaxrho}}(\cdot, \rho, k)` is the cdf of the normal maximum (equicorrelated case) with common correlation `\rho` and `k` groups (see :ref:`nmax_rho_cdf() `), and `f_{\chi}(\cdot, \nu)` is the pdf of the `\chi`-distribution with `\nu` degrees of freedom.
.. math:: \text{sf}(x) = 1 - \text{cdf}(x)
There is no known explicit expression for `\text{qtf}(q)` or `\text{isf}(q)`: These functions are computed with Newton iterations where the starting values are from the corresponding Boost functions (in double precision).
An example in Python
.. code-block:: pycon
>>> from mpfebnet import XReal
>>> XReal.Dunnett1Pdf(x, rho, k, nu)
XReal('5.2359877559829887307E-1')
>>> XReal.Dunnett1Pdf(x, rho, k, nu)
XReal('5.3518479027559984754E-1')
An example in Visual Basic
.. code-block:: pycon
>>> from mpfebnet import Gpr
>>> Gpr.Dunnett1Pdf(x, rho, k, nu)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Dunnett1Pdf(x, rho, k, nu)
Gpr('5.3518479027559984754E-1')
>>> from mpfebnet import XReal
>>> XReal.Dunnett1Cdf(x, rho, k, nu)
XReal('5.2359877559829887307E-1')
>>> XReal.Dunnett1Cdf(x, rho, k, nu)
XReal('5.3518479027559984754E-1')
>>> from mpfebnet import XReal
>>> XReal.Dunnett1Cdf(q, rho, k, nu)
XReal('5.2359877559829887307E-1')
>>> XReal.Dunnett1Cdf(q, rho, k, nu)
XReal('5.3518479027559984754E-1')
|newpage|
Dunnett `t`-distribution, 2-sided: pdf, cdf, qtf, boost class
-------------------------------------------------------------------------------
.. method:: math53lib.Dunnett2Pdf(x, rho, k, nu)
.. method:: math53lib.Dunnett2Cdf(x, rho, k, nu)
.. method:: math53lib.Dunnett2Qtf(q, rho, k, nu)
.. method:: math53lib.Dunnett2Dist(rho, k, nu)
Returns the pdf, cdf, qtf or boost class of a random variable `X`, following a 2-sided Dunnett `t`-distribution with common correlation `\rho`, `k \ge 2` groups (including control group), error degrees of freedom `\nu`, and the support interval `(0, +\infty)`, and `0 \le q \le 1`.
See also :cite:t:`Dunnett1955`, :cite:t:`Bechhofer1988`.
.. math:: \text{pdf}(x) = \int_{0}^\infty f_{\text{nmmrho}}(sx, \rho, k) \cdot s \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds
where `f_{\text{nmmrho}}(\cdot, \rho, k)` is the pdf of the normal maximum modulus (equicorrelated case) with common correlation `\rho` and `k` groups (see :ref:`nmm_rho_pdf() `), and `f_{\chi}(\cdot, \nu)` is the pdf of the `\chi`-distribution with `\nu` degrees of freedom.
.. math:: \text{cdf}(x) = \int_{0}^\infty F_{\text{nmmrho}}(sx, \rho, k) \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds
where `F_{\text{nmmrho}}(\cdot, \rho, k)` is the cdf of the normal maximum modulus (equicorrelated case) with common correlation `\rho` and `k` groups (see :ref:`nmm_rho_cdf() `), and `f_{\chi}(\cdot, \nu)` is the pdf of the `\chi`-distribution with `\nu` degrees of freedom.
.. math:: \text{sf}(x) = 1 - \text{cdf}(x)
There is no known explicit expression for `\text{qtf}(q)` or `\text{isf}(q)`: These functions are computed with Newton iterations where the starting values are from the corresponding Boost functions (in double precision).
An example in Python
.. code-block:: pycon
>>> from mpfebnet import XReal
>>> XReal.Dunnett2Pdf(x, rho, k, nu)
XReal('5.2359877559829887307E-1')
>>> XReal.Dunnett2Pdf(x, rho, k, nu)
XReal('5.3518479027559984754E-1')
An example in Visual Basic
.. code-block:: pycon
>>> from mpfebnet import Gpr
>>> Gpr.Dunnett2Pdf(x, rho, k, nu)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Dunnett2Pdf(x, rho, k, nu)
Gpr('5.3518479027559984754E-1')
>>> from mpfebnet import XReal
>>> XReal.Dunnett2Cdf(x, rho, k, nu)
XReal('5.2359877559829887307E-1')
>>> XReal.Dunnett2Cdf(x, rho, k, nu)
XReal('5.3518479027559984754E-1')
>>> from mpfebnet import XReal
>>> XReal.Dunnett2Qtf(q, rho, k, nu)
XReal('5.2359877559829887307E-1')
>>> XReal.Dunnett2Qtf(q, rho, k, nu)
XReal('5.3518479027559984754E-1')
|newpage|
Nair `t`-distribution: pdf, cdf, qtf, boost class
-------------------------------------------------------------------------------
.. method:: math53lib.NairPdf(x, rho, k, nu)
.. method:: math53lib.NairCdf(x, rho, k, nu)
.. method:: math53lib.NairQtf(q, rho, k, nu)
.. method:: math53lib.NairDist(rho, k, nu)
Returns the pdf, cdf, qtf or boost class of a random variable `X`, of a random variable `X`, following the Nair `t`-distribution with common correlation `\rho`, `k \ge 2` groups (including control group), error degrees of freedom `\nu`, and the support interval `(-\infty, +\infty)`, and `0 \le q \le 1`.
See also :cite:t:`Dunnett1955`, :cite:t:`Bechhofer1988`.
.. math:: \text{pdf}(x) = \int_{0}^\infty f_{\text{nmaxrho}}(sx, \rho, k) \cdot s \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds
where `f_{\text{nmaxrho}}(\cdot, \rho, k)` is the pdf of the normal maximum (equicorrelated case) with common correlation `\rho` and `k` groups (see :ref:`nmax_rho_pdf() `), and `f_{\chi}(\cdot, \nu)` is the pdf of the `\chi`-distribution with `\nu` degrees of freedom.
.. math:: \text{cdf}(x) = \int_{0}^\infty F_{\text{nmaxrho}}(sx, \rho, k) \cdot \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds
where `F_{\text{nmaxrho}}(\cdot, \rho, k)` is the cdf of the normal maximum (equicorrelated case) with common correlation `\rho` and `k` groups (see :ref:`nmax_rho_cdf() `), and `f_{\chi}(\cdot, \nu)` is the pdf of the `\chi`-distribution with `\nu` degrees of freedom.
.. math:: \text{sf}(x) = 1 - \text{cdf}(x)
An example in Python
.. code-block:: pycon
>>> from mpfebnet import XReal
>>> XReal.NairPdf(x, rho, k, nu)
XReal('5.2359877559829887307E-1')
>>> XReal.NairPdf(x, rho, k, nu)
XReal('5.3518479027559984754E-1')
An example in Visual Basic
.. code-block:: pycon
>>> from mpfebnet import Gpr
>>> Gpr.NairPdf(x, rho, k, nu)
Gpr('5.2359877559829887307E-1')
>>> Gpr.NairPdf(x, rho, k, nu)
Gpr('5.3518479027559984754E-1')
>>> from mpfebnet import XReal
>>> XReal.NairCdf(x, rho, k, nu)
XReal('5.2359877559829887307E-1')
>>> XReal.NairCdf(x, rho, k, nu)
XReal('5.3518479027559984754E-1')
>>> from mpfebnet import XReal
>>> XReal.NairQtf(q, rho, k, nu)
XReal('5.2359877559829887307E-1')
>>> XReal.NairQtf(q, rho, k, nu)
XReal('5.3518479027559984754E-1')
|newpage|
Halperin `t`-distribution: pdf, cdf, qtf, boost class
-------------------------------------------------------------------------------
.. method:: math53lib.HalperinPdf(x, rho, k, nu)
.. method:: math53lib.HalperinCdf(x, rho, k, nu)
.. method:: math53lib.HalperinQtf(q, rho, k, nu)
.. method:: math53lib.HalperinDist(rho, k, nu)
Returns the pdf, cdf, qtf or boost class of a random variable `X`, following a Halperin `t`-distribution with common correlation `\rho`, `k \ge 2` groups (including control group), error degrees of freedom `\nu`, and the support interval `(0, +\infty)`, and `0 \le q \le 1`.
See also :cite:t:`Dunnett1955`, :cite:t:`Bechhofer1988`.
.. math:: \text{pdf}(x) = \int_{0}^\infty f_{\text{nmmrho}}(sx, \rho, k) \cdot s \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds
where `f_{\text{nmmrho}}(\cdot, \rho, k)` is the pdf of the normal maximum modulus (equicorrelated case) with common correlation `\rho` and `k` groups (see :ref:`nmm_rho_pdf() `), and `f_{\chi}(\cdot, \nu)` is the pdf of the `\chi`-distribution with `\nu` degrees of freedom.
.. math:: \text{cdf}(x) = \int_{0}^\infty F_{\text{nmmrho}}(sx, \rho, k) \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds
where `F_{\text{nmmrho}}(\cdot, \rho, k)` is the cdf of the normal maximum modulus (equicorrelated case) with common correlation `\rho` and `k` groups (see :ref:`nmm_rho_cdf() `), and `f_{\chi}(\cdot, \nu)` is the pdf of the `\chi`-distribution with `\nu` degrees of freedom.
.. math:: \text{sf}(x) = 1 - \text{cdf}(x)
There is no known explicit expression for `\text{qtf}(q)` or `\text{isf}(q)`: These functions are computed with Newton iterations where the starting values are from the corresponding Boost functions (in double precision).
An example in Python
.. code-block:: pycon
>>> from mpfebnet import XReal
>>> XReal.HalperinPdf(x, rho, k, nu)
XReal('5.2359877559829887307E-1')
>>> XReal.HalperinPdf(x, rho, k, nu)
XReal('5.3518479027559984754E-1')
An example in Visual Basic
.. code-block:: pycon
>>> from mpfebnet import Gpr
>>> Gpr.HalperinPdf(x, rho, k, nu)
Gpr('5.2359877559829887307E-1')
>>> Gpr.HalperinPdf(x, rho, k, nu)
Gpr('5.3518479027559984754E-1')
>>> from mpfebnet import XReal
>>> XReal.HalperinCdf(x, rho, k, nu)
XReal('5.2359877559829887307E-1')
>>> XReal.HalperinCdf(x, rho, k, nu)
XReal('5.3518479027559984754E-1')
>>> from mpfebnet import XReal
>>> XReal.HalperinQtf(q, rho, k, nu)
XReal('5.2359877559829887307E-1')
>>> XReal.HalperinQtf(q, rho, k, nu)
XReal('5.3518479027559984754E-1')
|newpage|
Studentized range distribution: pdf, cdf, qtf, boost class
-------------------------------------------------------------------------------
.. method:: math53lib.StdrangePdf(x, k, nu)
.. method:: math53lib.StdrangeCdf(x, k, nu)
.. method:: math53lib.StdrangeQtf(q, k, nu)
.. method:: math53lib.StdrangeDist(k, nu)
Returns the pdf, cdf, qtf or boost class of a random variable `X`, following a studentized range distribution, with `k \ge 2` groups, `\nu` error degrees of freedom, and the support interval `(0, +\infty)`, and `0 \le q \le 1`.
See also Wikipedia :cite:p:`WikipediaDis60`, :cite:t:`Harter1960`, :cite:t:`RDis60`.
.. math:: \text{pdf}(x) = \int_{0}^\infty f_{\text{nrange}}(sx, k) \cdot s \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds
where `f_{\text{nrange}}(\cdot, k)` is the pdf of the normal range with `k` groups (see :ref:`nrange_pdf() `), and `f_{\chi}(\cdot, \nu)` is the pdf of the `\chi`-distribution with `\nu` degrees of freedom.
.. math:: \text{cdf}(x) = \int_{0}^\infty F_{\text{nrange}}(sx, k) \cdot \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds
where `F_{\text{nrange}}(\cdot, k)` is the cdf of the normal range with `k` groups (see :ref:`nrange_cdf() `), and `f_{\chi}(\cdot, \nu)` is the pdf of the `\chi`-distribution with `\nu` degrees of freedom.
.. math:: \text{sf}(x) = 1 - \text{cdf}(x)
There is no known explicit expression for `\text{qtf}(q)` or `\text{isf}(q)`: These functions are computed with Newton iterations where the starting values are from the corresponding Boost functions (in double precision).
An example in Python
.. code-block:: pycon
>>> from mpfebnet import XReal
>>> XReal.StdrangePdf(x, k, nu)
XReal('5.2359877559829887307E-1')
>>> XReal.StdrangePdf(x, k, nu)
XReal('5.3518479027559984754E-1')
An example in Visual Basic
.. code-block:: pycon
>>> from mpfebnet import Gpr
>>> Gpr.StdrangePdf(x, k, nu)
Gpr('5.2359877559829887307E-1')
>>> Gpr.StdrangePdf(x, k, nu)
Gpr('5.3518479027559984754E-1')
>>> from mpfebnet import XReal
>>> XReal.StdrangeCdf(x, k, nu)
XReal('5.2359877559829887307E-1')
>>> XReal.StdrangeCdf(x, k, nu)
XReal('5.3518479027559984754E-1')
>>> from mpfebnet import XReal
>>> XReal.StdrangeQtf(q, k, nu)
XReal('5.2359877559829887307E-1')
>>> XReal.StdrangeQtf(q, k, nu)
XReal('5.3518479027559984754E-1')