Boost: Map-Airy distribution

The Map-Airy distribution (or Airy distribution of the ’Map’-type) is a stable distribution (see Wikipedia [1330]) with the shape parameters \(\alpha = 3/2, \beta = 1\), which describes the probability distribution of the area under a Brownian excursion over a unit interval. For simplicity of numerical computation, this paper evaluates as follows assuming location parameter \(\mu = 0\), scale parameter \(c = 1 / \sqrt[3]{18}\), i.e. \(p(x) = p(x; \alpha = 3/2, \beta = 1, \mu = 0, c = 1 / \sqrt[3]{18})\). The Map-Airy 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/mapairy_dist.html

See also: https://mathworld.wolfram.com/Map-AiryDistribution.html

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

where Ctx is Math53 or CtxBoost.

Returns \(\text{pdf}(x)\), the value of the probability density function (Pdf) of the MapAiry distribution:

Boost:

\[\text{pdf}(x) = 2 e^{-2x^3 / 3} \left( x \text{Ai}(x^2) - \text{Ai}'(x^2)\right).\]

Wolfram:

\[\text{pdf}(x) = 2 e^{2x^3 / 3} \left( -x \text{Ai}(x^2) - \text{Ai}'(x^2)\right).\]

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.mapairy_cdf(x, a=0, b=1)

where Ctx is Math53 or CtxBoost.

Returns \(\text{cdf}(x)\), the value of the cumulative distribution function (Cdf) of the MapAiry 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.mapairy_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 MapAiry 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_mapairy(a=0, b=1)

Returns an dist_mapairy 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_mapairy.pdf(x)

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

dist_mapairy.cdf(x)

Returns \(\text{cdf}(x)\), the value of the cumulative distribution function of the Map-Airy distribution. See Ctx.mapairy_cdf for formulas and examples.

dist_mapairy.qtf(q)

Returns \(\text{qtf}(q)\), the value of the quantile function of the Map-Airy distribution. See Ctx.mapairy_qtf for formulas and examples.

dist_mapairy.sf(x)

Returns \(\text{sf}(x)\), the value of the survival function (Sf) of the Map-Airy 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_mapairy.isf(q)

Returns \(\text{isf}(q)\), the value of the inverse survival function (Isf) of the Map-Airy 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_mapairy.hf(x)

Returns \(\text{hazard}(x)\), the value of the hazard function (Hf) of the Map-Airy 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_mapairy.chf(x)

Returns \(\text{chf}(x)\), the value of the cumulative hazard function (Chf) of the Map-Airy 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_mapairy.mode()

Returns the mode of the Map-Airy 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_mapairy.median()

Returns the median of the Map-Airy 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_mapairy.mean()

Returns the mean (expected value) of the Map-Airy 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_mapairy.variance()

Returns the variance of the Map-Airy 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_mapairy.stdev()

Returns the standard deviation of the Map-Airy 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_mapairy.skewness()

Returns the skewness of the Map-Airy 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_mapairy.kurtosis()

Returns the ‘proper’ kurtosis (normalized fourth moment) of the Map-Airy 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_mapairy.kurtosis_excess()

Returns the kurtosis excess of the Map-Airy 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_mapairy.support_lower_endpoint()

Returns the support of the Map-Airy 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_mapairy.support_upper_endpoint()

Returns the support of the Map-Airy 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_mapairy.range_lower_endpoint()

Returns the valid range of the Map-Airy 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_mapairy.range_upper_endpoint()

Returns the valid range of the Map-Airy 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