Studentized maximum modulus distribution

class ctx.dist_smm(k, nu)

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

The studentized maximum modulus distribution with \(k \ge 1\) groups and \(\nu\) error degrees of freedom is a continuous distribution with the support interval \((-\infty, +\infty)\). See also Stoline and Ury [533], Hochberg and Tamhane [391], Narula [445], Bechhofer and Dunnett [31], and Hahn and Hendrickson [377].

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

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

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

dist_smm.pdf(x)

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

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

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.

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

dist_smm.cdf(x)

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

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

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.

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

dist_smm.sf(x)

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

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

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.

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

dist_smm.qtf(q)

Returns \(\text{qtf}_X(x)\), the quantile function function (qtf) of a random variable \(X\), following a studentized maximum modulus 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: ", smm(mu, sigma).qtf(q))
qtf: 6.3563523462564525615615615614561356E+00

dist_smm.isf(q)

Returns \(\text{isf}_X(q)\), the inverse survival function function (isf) of a random variable \(X\), following a studentized maximum modulus 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: ", smm(mu, sigma).isf(q))
6.3563523462564525615615615614561356E+00

dist_smm.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following a studentized 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: ", smm(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_smm.m_x(t)

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

dist_smm.k_x(t, k=0)

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

dist_smm.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a studentized maximum modulus 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: ", smm(mu, sigma).moments(k))
6.3563523462564525615615615614561356E+00

dist_smm.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following a 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: ", smm(mu, sigma).cumulants(k))
6.3563523462564525615615615614561356E+00