Normal maximum distribution, \(\rho_{ij, i \ne j} = \lambda_i \lambda_j\)

class ctx.dist_nmax_rhoij(rho, k)

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

The distribution of the maximum of \(k \ge 2\) correlated standard normal variates, with common correlation \(0 \le \rho \le 1\) is a continuous distribution with support interval \((-\infty, +\infty)\). See also Dunnett [304], Bechhofer and Dunnett [31], Grubbs [370], Bechhofer and Dunnett [31], Stoline and Ury [533], and Hahn and Hendrickson [377].

dist_nmax_rhoij.pdf(x)

Returns \(\text{pdf}_X(x)\), the probability density function (pdf) of a random variable \(X\), following the distribution of the maximum of \(k \ge 2\) correlated standard normal variables:

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

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

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

dist_nmax_rhoij.cdf(x)

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following the distribution of the maximum of \(k \ge 2\) correlated standard normal variables:

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

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

dist_nmax_rhoij.sf(x)

Returns \(\text{sf}_X(x)\), the survival function function (sf) of a random variable \(X\), following the distribution of the maximum of \(k \ge 2\) correlated standard normal variables:

\[\text{sf}_X(x) = 1 - \int_{-\infty}^\infty \left[\Phi \left(\frac{x + \sqrt{\vert \rho \vert} y} {\sqrt{1-\rho}} \right) \right]^n \phi(y) \mathrm{d} y\]

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

dist_nmax_rhoij.qtf(q)

Returns \(\text{qtf}_X(x)\), the quantile function function (qtf) of a random variable \(X\), following the distribution of the maximum of \(k \ge 2\) correlated standard normal variables:

There is no known explicit form for the quantile function \(\text{cdf}^{-1}_X(x)\): 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: ", mp_dunnett_t(mu, sigma).qtf(q))
qtf: 6.3563523462564525615615615614561356E+00

dist_nmax_rhoij.isf(q)

Returns \(\text{isf}_X(q)\), the inverse survival function function (isf) of a random variable \(X\), following the distribution of the maximum of \(k \ge 2\) correlated standard normal variables:

\[\text{isf}_X(q) = \text{qtf}_X(1-q) = \text{cdf}^{-1}_X(1-q).\]

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

dist_nmax_rhoij.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following the distribution of the maximum of \(k \ge 2\) correlated standard normal variables:

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

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

dist_nmax_rhoij.m_x(t)

Returns \(M_X(t)\), the moment generating function of a random variable \(X\), following the distribution of the maximum of \(k \ge 2\) correlated standard normal variables:

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

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

dist_nmax_rhoij.k_x(t, k=0)

Returns \(K_X(t)\), the cumulant generating function of a random variable \(X\), following the distribution of the maximum of \(k \ge 2\) correlated standard normal variables:

\[K_X(t) = \log\left(M_X(t)\right)\]

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

dist_nmax_rhoij.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following the distribution of the maximum of \(k \ge 2\) correlated standard normal variables.

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

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

dist_nmax_rhoij.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following the distribution of the maximum of \(k \ge 2\) correlated standard normal variables. The cumulants are calculated from the moments.

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