Normal maximum modulus distribution, \(\rho_{ij, i \ne j} = 0\)

class ctx.dist_nmm_0(k)

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

The normal maximum modulus distribution (the distribution of the maximum of the absolute value of \(k \ge 1\) independent standard normal variates) is a continuous probability distribution with the support interval \((0, +\infty)\).

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

\[Q_1=\frac{\text{max}X_j}{s}, \quad j=1,\ldots,k\]

follows a studentized maximum distribution with \(k\) and \(n\) degrees of freedom, and

\[Q_2=\frac{\text{max}|X_j|}{s}, \quad j=1,\ldots,k\]

follows a Studentized Maximum Modulus distribution with \(k\) and \(n\) degrees of freedom.

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

For tables see Bechhofer and Dunnett [31], Stoline and Ury [533], and Hahn and Hendrickson [377].

dist_nmm_0.pdf(x)

Returns \(\text{pdf}_X(x)\), the probability density function (pdf) of a random variable \(X\), following a normal maximum modulus distribution:

\[\text{pdf}_X(x) = f_{\text{nmm0}}(x, k) = 2k \cdot (2\Phi(x)-1)^{k-1} \cdot \phi(x), \quad x>0.\]

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

dist_nmm_0.cdf(x)

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following a normal maximum modulus distribution:

\[\text{cdf}_X(x) = F_{\text{nmm0}}(x, k) = \left[\Phi(x) - \Phi(-x)\right]^k = \left[2\Phi(x) - 1\right]^k, \quad x>0.\]

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

dist_nmm_0.sf(x)

Returns \(\text{sf}_X(x)\), the survival function function (sf) of a random variable \(X\), following a normal maximum modulus distribution:

\[\text{sf}_X(x) = 1 - \left[\Phi(x) - \Phi(-x)\right]^k = 1 - \left[2\Phi(x) - 1\right]^k, \quad x>0.\]

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

dist_nmm_0.qtf(q)

Returns \(\text{qtf}_X(x)\), the quantile function function (qtf) of a random variable \(X\), following a normal maximum modulus distribution:

\[\text{qtf}_X(x) = \Phi^{-1} \left(\tfrac{1}{2} + \tfrac{1}{2} q^{1/k} \right).\]

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

dist_nmm_0.isf(q)

Returns \(\text{isf}_X(q)\), the inverse survival function function (isf) of a random variable \(X\), following a normal maximum modulus distribution:

\[\text{isf}_X(x) = \Phi^{-1} \left(\tfrac{1}{2} + \tfrac{1}{2} (1-q)^{1/k} \right).\]

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

dist_nmm_0.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following a normal maximum modulus distribution:

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

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

dist_nmm_0.m_x(t)

Returns \(M_X(t)\), the moment generating function of a random variable \(X\), following a normal maximum modulus distribution:

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

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

dist_nmm_0.k_x(t, k=0)

Returns \(K_X(t)\), the cumulant generating function of a random variable \(X\), following a normal maximum modulus distribution:

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

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

dist_nmm_0.moments(k)

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

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

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

dist_nmm_0.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following a central Studentized Maximum Modulus distribution. The cumulants are calculated from the moments.

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