Distributions related to multiple comparisons of means

Normal maximum distribution, \(\rho_{ij, i \ne j} = \rho\): pdf, cdf, qtf, boost class

math53lib.NmaxRhoPdf(x, rho, k)

math53lib.NmaxRhoCdf(x, rho, k)

math53lib.NmaxRhoQtf(q, rho, k)

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 Dunnett [304], Bechhofer and Dunnett [31], Grubbs [370], Bechhofer and Dunnett [31], Stoline and Ury [533], and Hahn and Hendrickson [377].

\[\text{pdf}(x) = f_{\text{nmaxrho}}(x, \rho, k) = \frac{k}{b} \int_{-\infty}^\infty \left( \Phi(z_1) \right)^{k-1} \phi(z_1) \: \phi(y) \: dy,\]

\[\text{cdf}(x)= F_{\text{nmaxrho}}(x, \rho, k) = \int_{-\infty}^\infty \left( \Phi(z_1) \right)^{k} \phi(y) dy,\]

\[\text{sf}(x) = 1 - \int_{-\infty}^\infty \left( \Phi(z_1) \right)^{k} \phi(y) dy,\]

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

>>> from mpfebnet import XReal
>>> XReal.NmaxRhoPdf(x, rho, k)
XReal('5.2359877559829887307E-1')

An example in Visual Basic

>>> 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')

Normal maximum modulus distribution, \(\rho_{ij, i \ne j} = \rho\): pdf, cdf, qtf, boost class

math53lib.NmmRhoPdf(x, rho, k)

math53lib.NmmRhoCdf(x, rho, k)

math53lib.NmmRhoQtf(q, rho, k)

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 Dunnett [304], Bechhofer and Dunnett [31], Grubbs [370], Bechhofer and Dunnett [31], Stoline and Ury [533], and Hahn and Hendrickson [377].

\[\text{pdf}(x) = \frac{k}{b} \int_{-\infty}^\infty \left( \Phi(z_1)- \Phi(z_2) \right)^{k-1} (\phi(z_1)+\phi(z_1)) \: \phi(y) \: dy,\]

\[\text{cdf}(x)= F_{\text{nmaxrho}}(x, \rho, k) = \int_{-\infty}^\infty \left( \Phi(z_1) - \Phi(z_2) \right)^{k} \phi(y) dy,\]

\[\text{sf}(x) = 1 - \int_{-\infty}^\infty \left( \Phi(z_1) - \Phi(z_2) \right)^{k} \phi(y) dy,\]

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

>>> from mpfebnet import XReal
>>> XReal.NmmRhoPdf(x, rho, k)
XReal('5.2359877559829887307E-1')

An example in Visual Basic

>>> 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')

Normal maximum distribution, \(\rho_{ij, i \ne j} = \lambda_i \lambda_j\): pdf, cdf, qtf, boost class

math53lib.NmaxRhoijPdf(x, lambda_i, k)

math53lib.NmaxRhoijCdf(x, lambda_i, k)

math53lib.NmaxRhoijQtf(q, lambda_i, k)

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 Dunnett [304], Bechhofer and Dunnett [31], Grubbs [370], Bechhofer and Dunnett [31], Stoline and Ury [533], and Hahn and Hendrickson [377].

\[\text{pdf}(x) = f_{\text{nmaxrhoij}}(x, \rho, k) = \frac{k}{b} \int_{-\infty}^\infty \left( \Phi(z_1) \right)^{k-1} \phi(z_1) \: \phi(y) \: dy, \quad \text{where}\]

\[z_1 = \frac{x+a}{b}, \quad a = y \sqrt{|\rho|}, \quad b = \sqrt{1-\rho}, \rho = -\frac{1}{k-1}.\]

\[\text{cdf}(x)= F_{\text{nmaxrhoij}}(x, \rho, k) = \int_{-\infty}^\infty \left[\Phi \left(\frac{x + \sqrt{\vert \rho \vert} y} {\sqrt{1-\rho}} \right) \right]^n \phi(y) dy , \rho = -\frac{1}{k-1}.\]

\[\text{sf}(x) = 1 - \int_{-\infty}^\infty \left[\Phi \left(\frac{x + \sqrt{\vert \rho \vert} y} {\sqrt{1-\rho}} \right) \right]^n \phi(y) dy , \rho = -\frac{1}{k-1}.\]

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:

\[F_n(h;\rho_{ij}) = \int_{-\infty}^\infty \prod_{i=1}^n \left[\Phi \left(\frac{(a_i-\mu_i)/\sigma_i + \lambda_i z} {\sqrt{1-\lambda_i^2}} \right) \right] \phi(y) dz\]

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

>>> 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

>>> 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')

Normal maximum modulus distribution, \(\rho_{ij, i \ne j} = \lambda_i \lambda_j\): pdf, cdf, qtf, boost class

math53lib.NmmRhoijPdf(x, rho, k)

math53lib.NmmRhoijCdf(x, rho, k, cdf=True)

math53lib.NmmRhoijQtf(q, rho, k, qtf=True)

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 Dunnett [304], Bechhofer and Dunnett [31], Grubbs [370], Bechhofer and Dunnett [31], Stoline and Ury [533], and Hahn and Hendrickson [377].

\[\text{pdf}(x) = \frac{k}{b} \int_{-\infty}^\infty \left( \Phi(z_1)- \Phi(z_2) \right)^{k-1} (\phi(z_1)+\phi(z_1)) \: \phi(y) \: dy, \quad \text{where }\]

\[z_1 = \frac{x+a}{b}, z_2 = \frac{-x+a}{b}, \quad a = y \sqrt{|\rho|}, \quad b = \sqrt{1-\rho}, \rho = -\frac{1}{k-1}.\]

\[\text{cdf}(x) = \int_{-\infty}^\infty \left[\Phi \left(\frac{x + \sqrt{\vert \rho \vert} y} {\sqrt{1-\rho}} \right) - \Phi \left(\frac{-x + \sqrt{\vert \rho \vert} y} {\sqrt{1-\rho}} \right) \right]^k \phi(y) dy , \rho = -\frac{1}{k-1}.\]

\[\text{sf}(x) = 1 - \int_{-\infty}^\infty \left[\Phi \left(\frac{x + \sqrt{\vert \rho \vert} y} {\sqrt{1-\rho}} \right) - \Phi \left(\frac{-x + \sqrt{\vert \rho \vert} y} {\sqrt{1-\rho}} \right) \right]^k \phi(y) dy , \rho = -\frac{1}{k-1}.\]

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:

\[F_n(h;\rho_{ij}) = \int_{-\infty}^\infty \prod_{i=1}^n \left[\Phi \left(\frac{(a_i-\mu_i)/\sigma_i + \lambda_i z} {\sqrt{1-\lambda_i^2}} \right) - \Phi \left(\frac{(b_i-\mu_i)/\sigma_i + \lambda_i z} {\sqrt{1-\lambda_i^2}} \right) \right] \phi(y) dz\]

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

>>> 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

>>> 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')

Normal range distribution: pdf, cdf, qtf, boost class

math53lib.NrangePdf(x, k)

math53lib.NrangeCdf(n, x)

math53lib.NrangeQtf(q, k)

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 [1310], Harter [381], Team [561].

The density in the central case is given by

\[\text{pdf}(x) = k(k-1) \int_{-\infty}^\infty \left( \Phi(y) - \Phi(y-x) \right)^{k-2} \phi(y) \phi(y-x) dy\]

We have

\[\text{cdf}(x) = k \int_{-\infty}^\infty \phi(y) \left( \Phi(y) - \Phi(y-x) \right)^{k-1} dy\]

\[\text{sf}(x) = 1 - k \int_{-\infty}^\infty \phi(y) \left( \Phi(y) - \Phi(y-x) \right)^{k-1} dy\]

In the noncentral case, this becomes

\[\text{cdf}(x) = F_sr \left(m,n,x\right) = \sum_{i=1}^k \int_{-\infty}^\infty \phi(y_i - \mu_i) \left( \prod_{i=1, j \neq i}^k (\Phi(y_i - \mu_i) - \Phi(y_i - \mu_i -x) \right) dy_i.\]

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

>>> from mpfebnet import XReal
>>> XReal.NrangePdf(x, k)
XReal('5.2359877559829887307E-1')
>>> XReal.NrangePdfxq, k)
XReal('5.3518479027559984754E-1')

An example in Visual Basic

>>> 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')

Studentized maximum distribution: pdf, cdf, qtf, boost class

math53lib.SmaxPdf(x, k, nu)

math53lib.SmaxCdf(x, k, nu)

math53lib.SmaxQtf(q, k, nu)

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 Stoline and Ury [533], Hochberg and Tamhane [391], Narula [445].

\[\text{pdf}(x) = \int_{0}^\infty f_{\text{nnax}}(sx, k) \cdot s \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds\]

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.

\[\text{cdf}(x) = \int_{0}^\infty F_{\text{nmax}}(sx, k) \cdot \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds\]

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.

\[\text{sf}(x) = 1 - \text{cdf}(x)\]

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

>>> 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

>>> 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')

Studentized maximum modulus distribution: pdf, cdf, qtf, boost class

math53lib.SmmPdf(x, k, nu)

math53lib.SmmCdf(x, k, nu)

math53lib.SmmQtf(q, k, nu)

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 Stoline and Ury [533], Hochberg and Tamhane [391], Narula [445].

\[\text{pdf}(x) = \int_{0}^\infty f_{\text{nmm}}(sx, k) \cdot s \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds\]

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.

\[\text{cdf}(x) = \int_{0}^\infty F_{\text{nmm}}(sx, k) \cdot \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds\]

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.

\[\text{sf}(x) = 1 - \text{cdf}(x)\]

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

>>> 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

>>> 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')

Dunnett \(t\)-distribution, 1-sided: pdf, cdf, qtf, boost class

math53lib.Dunnett1Pdf(x, rho, k, nu)

math53lib.Dunnett1Cdf(x, rho, k, nu)

math53lib.Dunnett1Qtf(q, rho, k, nu)

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 Dunnett [304], Bechhofer and Dunnett [31].

\[\text{pdf}(x) = \int_{0}^\infty f_{\text{nmaxrho}}(sx, \rho, k) \cdot s \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds\]

where \(f_{\text{nmaxrho}}(\cdot, \rho, k)\) is the pdf of the normal maximum (equicorrelated case) with common correlation \(\rho\) and \(k\) groups (see nmax_rho_pdf()), and \(f_{\chi}(\cdot, \nu)\) is the pdf of the \(\chi\)-distribution with \(\nu\) degrees of freedom.

\[\text{cdf}(x) = \int_{0}^\infty F_{\text{nmaxrho}}(sx, \rho, k) \cdot \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds\]

where \(F_{\text{nmaxrho}}(\cdot, \rho, k)\) is the cdf of the normal maximum (equicorrelated case) with common correlation \(\rho\) and \(k\) groups (see nmax_rho_cdf()), and \(f_{\chi}(\cdot, \nu)\) is the pdf of the \(\chi\)-distribution with \(\nu\) degrees of freedom.

\[\text{sf}(x) = 1 - \text{cdf}(x)\]

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

>>> 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

>>> 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')

Dunnett \(t\)-distribution, 2-sided: pdf, cdf, qtf, boost class

math53lib.Dunnett2Pdf(x, rho, k, nu)

math53lib.Dunnett2Cdf(x, rho, k, nu)

math53lib.Dunnett2Qtf(q, rho, k, nu)

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 Dunnett [304], Bechhofer and Dunnett [31].

\[\text{pdf}(x) = \int_{0}^\infty f_{\text{nmmrho}}(sx, \rho, k) \cdot s \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds\]

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 nmm_rho_pdf()), and \(f_{\chi}(\cdot, \nu)\) is the pdf of the \(\chi\)-distribution with \(\nu\) degrees of freedom.

\[\text{cdf}(x) = \int_{0}^\infty F_{\text{nmmrho}}(sx, \rho, k) \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds\]

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 nmm_rho_cdf()), and \(f_{\chi}(\cdot, \nu)\) is the pdf of the \(\chi\)-distribution with \(\nu\) degrees of freedom.

\[\text{sf}(x) = 1 - \text{cdf}(x)\]

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

>>> 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

>>> 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')

Nair \(t\)-distribution: pdf, cdf, qtf, boost class

math53lib.NairPdf(x, rho, k, nu)

math53lib.NairCdf(x, rho, k, nu)

math53lib.NairQtf(q, rho, k, nu)

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 Dunnett [304], Bechhofer and Dunnett [31].

\[\text{pdf}(x) = \int_{0}^\infty f_{\text{nmaxrho}}(sx, \rho, k) \cdot s \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds\]

where \(f_{\text{nmaxrho}}(\cdot, \rho, k)\) is the pdf of the normal maximum (equicorrelated case) with common correlation \(\rho\) and \(k\) groups (see nmax_rho_pdf()), and \(f_{\chi}(\cdot, \nu)\) is the pdf of the \(\chi\)-distribution with \(\nu\) degrees of freedom.

\[\text{cdf}(x) = \int_{0}^\infty F_{\text{nmaxrho}}(sx, \rho, k) \cdot \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds\]

where \(F_{\text{nmaxrho}}(\cdot, \rho, k)\) is the cdf of the normal maximum (equicorrelated case) with common correlation \(\rho\) and \(k\) groups (see nmax_rho_cdf()), and \(f_{\chi}(\cdot, \nu)\) is the pdf of the \(\chi\)-distribution with \(\nu\) degrees of freedom.

\[\text{sf}(x) = 1 - \text{cdf}(x)\]

An example in Python

>>> 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

>>> 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')

Halperin \(t\)-distribution: pdf, cdf, qtf, boost class

math53lib.HalperinPdf(x, rho, k, nu)

math53lib.HalperinCdf(x, rho, k, nu)

math53lib.HalperinQtf(q, rho, k, nu)

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 Dunnett [304], Bechhofer and Dunnett [31].

\[\text{pdf}(x) = \int_{0}^\infty f_{\text{nmmrho}}(sx, \rho, k) \cdot s \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds\]

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 nmm_rho_pdf()), and \(f_{\chi}(\cdot, \nu)\) is the pdf of the \(\chi\)-distribution with \(\nu\) degrees of freedom.

\[\text{cdf}(x) = \int_{0}^\infty F_{\text{nmmrho}}(sx, \rho, k) \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds\]

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 nmm_rho_cdf()), and \(f_{\chi}(\cdot, \nu)\) is the pdf of the \(\chi\)-distribution with \(\nu\) degrees of freedom.

\[\text{sf}(x) = 1 - \text{cdf}(x)\]

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

>>> 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

>>> 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')

Studentized range distribution: pdf, cdf, qtf, boost class

math53lib.StdrangePdf(x, k, nu)

math53lib.StdrangeCdf(x, k, nu)

math53lib.StdrangeQtf(q, k, nu)

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 [1310], Harter [381], Team [561].

\[\text{pdf}(x) = \int_{0}^\infty f_{\text{nrange}}(sx, k) \cdot s \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds\]

where \(f_{\text{nrange}}(\cdot, k)\) is the pdf of the normal range with \(k\) groups (see nrange_pdf()), and \(f_{\chi}(\cdot, \nu)\) is the pdf of the \(\chi\)-distribution with \(\nu\) degrees of freedom.

\[\text{cdf}(x) = \int_{0}^\infty F_{\text{nrange}}(sx, k) \cdot \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: ds\]

where \(F_{\text{nrange}}(\cdot, k)\) is the cdf of the normal range with \(k\) groups (see nrange_cdf()), and \(f_{\chi}(\cdot, \nu)\) is the pdf of the \(\chi\)-distribution with \(\nu\) degrees of freedom.

\[\text{sf}(x) = 1 - \text{cdf}(x)\]

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

>>> 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

>>> 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')