Boost: Holtsmark distribution

The Holtsmark distribution is a stable distribution (see Wikipedia [1330]) with the shape parameters \(\alpha=3/2, \beta=0\). The Holtsmark distribution is linear with respect to the location parameter \(\mu\) and scale parameter \(c\).

\[p(x_1; \mu_1, c_1) = p \left(x := \frac{x_1 - \mu_1}{c_1}; \mu:=0, c := 1 \right) \cdot \frac{1}{c_1}.\]

The support interval is \((-\infty,+\infty)\).

!!! The following references need to be updated: !!!

See also: https://www.boost.org/doc/libs/1_89_0/libs/math/doc/html/math_toolkit/dist_ref/dists/holtsmark_dist.html

See also: https://en.wikipedia.org/wiki/Holtsmark_distribution

Ctx.holtsmark_pdf(x, m, n)

where Ctx is Math53 or CtxBoost.

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

\[\text{pdf}_X(x) =\frac{1}{2\pi} \int_{-\infty}^{\infty} \exp \left ( i t \mu - |c t|^{3/2} \right ) e^{i x t} \mathrm{d}t,\]

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("FisherFPdf(x, a, b): ", FisherFPdf(x, a, b))
>>> print ("dist_fisher_f(a, b).pdf(x): ", dist_fisher_f(a, b).pdf(x))
6.3563523462564525615615615614561356E+00

Ctx.holtsmark_cdf(x, m, n)

where Ctx is Math53 or CtxBoost.

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following an Holtsmark distribution.

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("FisherFCdf(x, a, b): ", FisherFCdf(x, a, b))
>>> print ("dist_fisher_f(a, b).cdf(x): ", dist_fisher_f(a, b).cdf(x))
6.3563523462564525615615615614561356E+00

Ctx.holtsmark_qtf(q, m, n)

where Ctx is Math53 or CtxBoost.

Returns \(\text{qtf}(q)\), the value of the quantile function (Qtf) of the Holtsmark distribution.

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; q = 0.6;
>>> print ("FisherFQtf(q, a, b): ", FisherFQtf(q, a, b))
>>> print ("dist_fisher_f(a, b).qtf(q): ", dist_fisher_f(a, b).qtf(q))
6.3563523462564525615615615614561356E+00

CtxBoost.dist_holtsmark(m, n)

Returns an dist_holtsmark object, which gives access to the functions descibed below:

>>> from mpfebnet import SReal, FReal, XReal, QReal, CReal, OReal
>>> a = 0; b = 1;
>>> Ctx = SReal
>>> dist_fisher_f = Ctx.dist_fisher_f(a, b)
>>> print ("Dist.qtf(q=0.5): ", Dist.qtf(q=0.5))
6.3563523462564525615615615614561356E+00

dist_holtsmark.pdf(x)

Returns \(\text{pdf}(x)\), the value of the probability density function of the Holtsmark distribution. See Ctx.holtsmark_pdf for formulas and examples.

dist_holtsmark.cdf(x)

Returns \(\text{cdf}(x)\), the value of the cumulative distribution function of the Holtsmark distribution. See Ctx.holtsmark_cdf for formulas and examples.

dist_holtsmark.qtf(q)

Returns \(\text{qtf}(q)\), the value of the quantile function of the Holtsmark distribution. See Ctx.holtsmark_qtf for formulas and examples.

dist_holtsmark.sf(x)

Returns \(\text{sf}(x)\), the value of the survival function (Sf) of the Holtsmark distribution.

\[\begin{split}\text{sf}(x) =\begin{cases} \text{ibeta}(n/2, m/2, n/(n+mx)), & mx > n,\\ \text{ibetac}(m/2, n/2, mx/(n+mx)) & mx \le n. \end{cases}\end{split}\]

Here \(\text{ibeta}(\cdot)\) denotes the real normalised incomplete beta function (RealIBeta), and \(\text{ibetac}(\cdot)\) denotes the real normalised complementary incomplete beta function (RealIBetac).

>>> # continued from above
>>> print ("Dist.sf(x=0.5): ", Dist.qtf(x=0.5))
6.3563523462564525615615615614561356E+00

dist_holtsmark.isf(q)

Returns \(\text{isf}(q)\), the value of the inverse survival function (Isf) of the Holtsmark distribution.

\[\text{isf}(q) = \frac{nx}{m(1-x)}, \quad \text{where } x = \mathrm{ibetac\_inv}(m/2, n/2, q).\]

Here \(\mathrm{ibetac\_inv}(\cdot)\) denotes the inverse of the real normalised complementary incomplete beta function (RealIBetacInv).

>>> # continued from above
>>> print ("Dist.isf(x=0.5): ", Dist.isf(x=0.5))
6.3563523462564525615615615614561356E+00

dist_holtsmark.hf(x)

Returns \(\text{hazard}(x)\), the value of the hazard function (Hf) of the Holtsmark distribution.

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

dist_holtsmark.chf(x)

Returns \(\text{chf}(x)\), the value of the cumulative hazard function (Chf) of the Holtsmark distribution.

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

dist_holtsmark.mode()

Returns the mode of the Holtsmark distribution. Since there is not one unique mode, Nan is returned.

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

dist_holtsmark.median()

Returns the median of the Holtsmark distribution. Calculated as \(\displaystyle \tfrac{1}{2}(a+b)\).

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

dist_holtsmark.mean()

Returns the mean (expected value) of the Holtsmark distribution. Calculated as \(\displaystyle \tfrac{1}{2}(a+b)\).

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

dist_holtsmark.variance()

Returns the variance of the Holtsmark distribution. Calculated as \(\displaystyle \tfrac{1}{8}(b-a)^2\).

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

dist_holtsmark.stdev()

Returns the standard deviation of the Holtsmark distribution. Calculated as \(\displaystyle \sqrt{ \tfrac{1}{8}(b-a)^2}\).

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

dist_holtsmark.skewness()

Returns the skewness of the Holtsmark distribution. Calculated as \(\displaystyle 0\).

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

dist_holtsmark.kurtosis()

Returns the ‘proper’ kurtosis (normalized fourth moment) of the Holtsmark distribution. Calculated as \(\displaystyle 3/2\).

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

dist_holtsmark.kurtosis_excess()

Returns the kurtosis excess of the Holtsmark distribution. Calculated as \(\displaystyle -3/2\).

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

dist_holtsmark.support_lower_endpoint()

Returns the support of the Holtsmark distribution as a tuple (left, right).

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

dist_holtsmark.support_upper_endpoint()

Returns the support of the Holtsmark distribution as a tuple (left, right).

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

dist_holtsmark.range_lower_endpoint()

Returns the valid range of the Holtsmark distribution as a tuple (left, right).

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

dist_holtsmark.range_upper_endpoint()

Returns the valid range of the Holtsmark distribution as a tuple (left, right).

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