Nelson’s \(h\)-distribution

class ctx.dist_nelson_h(rho, k, nu)

where ctx is fpm, mpm, ipm, dec, gmp or apm.

The Halperin \(t\)-distribution with common correlation \(\rho\), \(k \ge 2\) groups (including control group) and error degrees of freedom \(\nu\) is a continuous distribution with the support interval \((0, +\infty)\). See also Dunnett [304], Bechhofer and Dunnett [31].

Let \(X_1,\ldots,X_k\) be a random sample of size \(k\) from a \(\mathcal{N}(0,\sigma^2)\) distribution. Let \(s^2\) be an independent mean square estimate of \(\sigma\) with \(n\) degrees of freedom. Then

\[Q=\frac{\text{max}|X_1 - X_j|}{s}, \quad 1<j<k\]

has a Halperin \(|t|\)-distribution with \(k\) and \(n\) degrees of freedom.

For Dunnett’s test:

\[\lambda_i=\frac{1}{\sqrt{1+n_0/n_i}}\]

dist_nelson_h.pdf(x)

Returns \(\text{pdf}_X(x)\), the probability density function (pdf) of a random variable \(X\), following a Halperin t-distribution:

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

where \(f_{\mathrm{nmm\_corr}}(\cdot, \rho, k)\) is the pdf of the normal maximum modulus (equicorrelated case) with common correlation \(\rho\) and \(k\) groups, and \(f_{\chi}(\cdot, \nu)\) is the pdf of the \(\chi\)-distribution with \(\nu\) degrees of freedom.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("pdf: ", nelson_h(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_nelson_h.cdf(x)

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following a Halperin t-distribution:

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

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

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("cdf: ", nelson_h(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_nelson_h.sf(x)

Returns \(\text{sf}_X(x)\), the survival function function (sf) of a random variable \(X\), following a Halperin t-distribution:

\[\text{sf}_X(x) = 1 - \int_{0}^\infty F_{\mathrm{nmm\_corr}}(sx, \rho, k) \sqrt{\nu} \cdot f_{\chi} \left(s \sqrt{\nu}, \nu \right) \: \mathrm{d} s\]

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

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print (" sf: ", nelson_h(mu, sigma).pdf(x))
sf: 6.3563523462564525615615615614561356E-20

dist_nelson_h.qtf(q)

Returns \(\text{qtf}_X(x)\), the quantile function function (qtf) of a random variable \(X\), following a Halperin t-distribution:

There is no known explicit form for the quantile function \(\text{qtf}_X(x)\): It is computed using Newton iterations with starting values from a central \(F\) approximation.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("qtf: ", nelson_h(mu, sigma).qtf(q))
qtf: 6.3563523462564525615615615614561356E+00

dist_nelson_h.isf(q)

Returns \(\text{isf}_X(q)\), the inverse survival function function (isf) of a random variable \(X\), following a Halperin t-distribution:

There is no known explicit form for the quantile function \(\text{isf}_X(x)\): It is computed using Newton iterations with starting values from a central \(F\) approximation.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("isf: ", nelson_h(mu, sigma).isf(q))
6.3563523462564525615615615614561356E+00

dist_nelson_h.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following a Halperin t-distribution:

\[C_X(t) = \int_{0}^{\infty} e^{i tx} \text{pdf}_X(x) \mathrm{d} x\]

where \(U(\cdot)\) denotes the confluent hypergeometric function of the second kind.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3;
>>> print ("c_x: ", nelson_h(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_nelson_h.m_x(t)

Returns NaN, since the moment generating function does not exist.

dist_nelson_h.k_x(t, k=0)

Returns NaN, since the cumulant generating function does not exist.

dist_nelson_h.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a Halperin t-distribution. The rth moments only exists for \(n_2 > 2r\).

\[\mu'_X(r) = E(X^r) = \int_{0}^{\infty} x^r \text{pdf}_X(x) \mathrm{d} x\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", nelson_h(mu, sigma).moments(k))
6.3563523462564525615615615614561356E+00

dist_nelson_h.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following a Halperin t-distribution. The cumulants are calculated from the moments.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", nelson_h(mu, sigma).cumulants(k))
6.3563523462564525615615615614561356E+00