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

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

class ctx.dist_nmax_0(k)#

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

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

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_nmax_0.pdf(x)#

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

\[\text{pdf}_X(x) = f_{\text{nmax0}}(x, k) = k \cdot \Phi(x)^{k-1} \cdot \phi(x).\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("pdf: ", mp_smm(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_nmax_0.cdf(x)#

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

\[\text{cdf}_X(x) = F_{\text{nmax0}}(x, k) = \left[\Phi(x)\right]^k.\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("cdf: ", mp_smm(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_nmax_0.sf(x)#

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

\[\text{sf}_X(x) = 1 - \left[\Phi(x)\right]^k.\]
>>> 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_nmax_0.qtf(q)#

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

\[\text{qtf}_X(x) = \Phi^{-1} \left( q^{1/k} \right).\]
>>> 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_nmax_0.isf(q)#

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

\[\text{isf}_X(x) = \Phi^{-1} \left( (1-q)^{1/k} \right).\]
>>> 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_nmax_0.c_x(t)#

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following a normal maximum 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_nmax_0.m_x(t)#

Returns \(M_X(t)\), the moment generating function of a random variable \(X\), following a normal maximum 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_nmax_0.k_x(t, k=0)#

Returns \(K_X(t)\), the cumulant generating function of a random variable \(X\), following a normal maximum 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_nmax_0.moments(k)#

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a normal maximum 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_nmax_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
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