.. |newpage| raw:: latex \newpage .. |cr| raw:: latex \hspace{0.0mm} |newpage| Boost: Holtsmark distribution ------------------------------------------------------------------------------- The Holtsmark distribution is a stable distribution (see Wikipedia :cite:p:`WikipediaDis94`) 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`. .. math:: 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 |cr| .. _Ctx_Holtsmark_Pdf: .. method:: 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 .. math:: \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: .. code-block:: pycon >>> 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 |cr| .. _Ctx_Holtsmark_Cdf: .. method:: 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: .. code-block:: pycon >>> 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 |cr| .. _Ctx_Holtsmark_Qtf: .. method:: Ctx.holtsmark_qtf(q, m, n) where ``Ctx`` is ``Math53`` or ``CtxBoost``. Returns `\text{qtf}(q)`, the value of the quantile function (:ref:`Qtf `) of the Holtsmark distribution. The following example shows both forms of the syntax: .. code-block:: pycon >>> 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 |cr| .. method:: CtxBoost.dist_holtsmark(m, n) Returns an ``dist_holtsmark`` object, which gives access to the functions descibed below: .. code-block:: pycon >>> 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 .. method:: dist_holtsmark.pdf(x) Returns `\text{pdf}(x)`, the value of the probability density function of the Holtsmark distribution. See :ref:`Ctx.holtsmark_pdf ` for formulas and examples. .. method:: dist_holtsmark.cdf(x) Returns `\text{cdf}(x)`, the value of the cumulative distribution function of the Holtsmark distribution. See :ref:`Ctx.holtsmark_cdf ` for formulas and examples. .. method:: dist_holtsmark.qtf(q) Returns `\text{qtf}(q)`, the value of the quantile function of the Holtsmark distribution. See :ref:`Ctx.holtsmark_qtf ` for formulas and examples. |cr| .. method:: dist_holtsmark.sf(x) Returns `\text{sf}(x)`, the value of the survival function (:ref:`Sf `) of the Holtsmark distribution. .. math:: \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} Here `\text{ibeta}(\cdot)` denotes the real normalised incomplete beta function (:ref:`RealIBeta `), and `\text{ibetac}(\cdot)` denotes the real normalised complementary incomplete beta function (:ref:`RealIBetac `). .. code-block:: pycon >>> # continued from above >>> print ("Dist.sf(x=0.5): ", Dist.qtf(x=0.5)) 6.3563523462564525615615615614561356E+00 .. method:: dist_holtsmark.isf(q) Returns `\text{isf}(q)`, the value of the inverse survival function (:ref:`Isf `) of the Holtsmark distribution. .. math:: \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 (:ref:`RealIBetacInv `). .. code-block:: pycon >>> # continued from above >>> print ("Dist.isf(x=0.5): ", Dist.isf(x=0.5)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_holtsmark.hf(x) Returns `\text{hazard}(x)`, the value of the hazard function (:ref:`Hf `) of the Holtsmark distribution. .. code-block:: pycon >>> 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 |cr| .. method:: dist_holtsmark.chf(x) Returns `\text{chf}(x)`, the value of the cumulative hazard function (:ref:`Chf `) of the Holtsmark distribution. .. code-block:: pycon >>> 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 |cr| .. method:: dist_holtsmark.mode() Returns the :ref:`mode ` of the Holtsmark distribution. Since there is not one unique mode, ``Nan`` is returned. .. code-block:: pycon >>> 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 |cr| .. method:: dist_holtsmark.median() Returns the :ref:`median ` of the Holtsmark distribution. Calculated as `\displaystyle \tfrac{1}{2}(a+b)`. .. code-block:: pycon >>> 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 |cr| .. method:: dist_holtsmark.mean() Returns the :ref:`mean ` (expected value) of the Holtsmark distribution. Calculated as `\displaystyle \tfrac{1}{2}(a+b)`. .. code-block:: pycon >>> 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 |cr| .. method:: dist_holtsmark.variance() Returns the :ref:`variance ` of the Holtsmark distribution. Calculated as `\displaystyle \tfrac{1}{8}(b-a)^2`. .. code-block:: pycon >>> 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 |cr| .. method:: dist_holtsmark.stdev() Returns the :ref:`standard deviation ` of the Holtsmark distribution. Calculated as `\displaystyle \sqrt{ \tfrac{1}{8}(b-a)^2}`. .. code-block:: pycon >>> 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 |cr| .. method:: dist_holtsmark.skewness() Returns the :ref:`skewness ` of the Holtsmark distribution. Calculated as `\displaystyle 0`. .. code-block:: pycon >>> 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 |cr| .. method:: dist_holtsmark.kurtosis() Returns the 'proper' :ref:`kurtosis ` (normalized fourth moment) of the Holtsmark distribution. Calculated as `\displaystyle 3/2`. .. code-block:: pycon >>> 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 |cr| .. method:: dist_holtsmark.kurtosis_excess() Returns the :ref:`kurtosis excess ` of the Holtsmark distribution. Calculated as `\displaystyle -3/2`. .. code-block:: pycon >>> 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 |cr| .. method:: dist_holtsmark.support_lower_endpoint() Returns the :ref:`support ` of the Holtsmark distribution as a tuple (left, right). .. code-block:: pycon >>> 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 |cr| .. method:: dist_holtsmark.support_upper_endpoint() Returns the :ref:`support ` of the Holtsmark distribution as a tuple (left, right). .. code-block:: pycon >>> 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 |cr| .. method:: dist_holtsmark.range_lower_endpoint() Returns the valid :ref:`range ` of the Holtsmark distribution as a tuple (left, right). .. code-block:: pycon >>> 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 |cr| .. method:: dist_holtsmark.range_upper_endpoint() Returns the valid :ref:`range ` of the Holtsmark distribution as a tuple (left, right). .. code-block:: pycon >>> 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