Boost: Saspoint5 distribution

The SaS point5 distribution is a stable distribution (see Wikipedia [1330]) with the shape parameters \(\alpha=1/2, \beta=0\). The Saspoint5 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/saspoint5_dist.html

See also: https://en.wikipedia.org/wiki/Stable_distribution#Other_analytic_cases

Ctx.saspoint5_pdf(x, a=0, b=1)

where Ctx is Math53 or CtxBoost.

Returns \(\text{pdf}_X(x)\), the probability density function (pdf) of a random variable \(X\), following an Saspoint5 distribution. Let \(S(x)\) and \(C(x)\) denote the Fresnel integrals. Then:

\[\text{pdf}_X(x) ={\frac {1}{\sqrt {2\pi |x|^{3}}}}\left(\sin \left({\tfrac {1}{4|x|}}\right)\left[{\frac {1}{2}}-S\left({\tfrac {1}{\sqrt {2\pi |x|}}}\right)\right]+\cos \left({\tfrac {1}{4|x|}}\right)\left[{\frac {1}{2}}-C\left({\tfrac {1}{\sqrt {2\pi |x|}}}\right)\right]\right).\]

The following example shows both forms of the syntax:

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

Ctx.saspoint5_cdf(x, a=0, b=1)

where Ctx is Math53 or CtxBoost.

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following an Saspoint5 distribution.

The following example shows both forms of the syntax:

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

Ctx.saspoint5_qtf(q, a=0, b=1)

where Ctx is Math53 or CtxBoost.

Returns \(\text{qtf}(q)\), the value of the quantile function (Qtf) of the Saspoint5 distribution:

The following example shows both forms of the syntax:

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

CtxBoost.dist_saspoint5(a=0, b=1)

Returns an dist_saspoint5 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_student_t = Ctx.dist_student_t(a, b)
>>> print ("Dist.qtf(q=0.5): ", Dist.qtf(q=0.5))
6.3563523462564525615615615614561356E+00

dist_saspoint5.pdf(x)

Returns \(\text{pdf}(x)\), the value of the probability density function of the Saspoint5 distribution. See Ctx.saspoint5_pdf for formulas and examples.

dist_saspoint5.cdf(x)

Returns \(\text{cdf}(x)\), the value of the cumulative distribution function of the Saspoint5 distribution. See Ctx.saspoint5_cdf for formulas and examples.

dist_saspoint5.qtf(q)

Returns \(\text{qtf}(q)\), the value of the quantile function of the Saspoint5 distribution. See Ctx.saspoint5_qtf for formulas and examples.

dist_saspoint5.sf(x)

Returns \(\text{sf}(x)\), the value of the survival function (Sf) of the Saspoint5 distribution.

\[\text{sf}(x) = \text{cdf}_X(-x)\]

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

dist_saspoint5.isf(q)

Returns \(\text{isf}(q)\), the value of the inverse survival function (Isf) of the Saspoint5 distribution.

\[\text{isf}(q) = -\text{qtf}(q).\]

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

dist_saspoint5.hf(x)

Returns \(\text{hazard}(x)\), the value of the hazard function (Hf) of the Saspoint5 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_saspoint5.chf(x)

Returns \(\text{chf}(x)\), the value of the cumulative hazard function (Chf) of the Saspoint5 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_saspoint5.mode()

Returns the mode of the Saspoint5 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_saspoint5.median()

Returns the median of the Saspoint5 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_saspoint5.mean()

Returns the mean (expected value) of the Saspoint5 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_saspoint5.variance()

Returns the variance of the Saspoint5 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_saspoint5.stdev()

Returns the standard deviation of the Saspoint5 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_saspoint5.skewness()

Returns the skewness of the Saspoint5 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_saspoint5.kurtosis()

Returns the ‘proper’ kurtosis (normalized fourth moment) of the Saspoint5 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_saspoint5.kurtosis_excess()

Returns the kurtosis excess of the Saspoint5 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_saspoint5.support_lower_endpoint()

Returns the support of the Saspoint5 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_saspoint5.support_upper_endpoint()

Returns the support of the Saspoint5 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_saspoint5.range_lower_endpoint()

Returns the valid range of the Saspoint5 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_saspoint5.range_upper_endpoint()

Returns the valid range of the Saspoint5 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