.. |newpage| raw:: latex \newpage .. |cr| raw:: latex \hspace{0.0mm} Boost: Kolmogorov-Smirnov distribution (limiting form) ------------------------------------------------------------------------------- Returns the pdf, cdf, qtf or boost class of a random variable `X`, following the limiting form of the Kolmogorov distribution with parameters `n > 0` and the support interval `[0, \infty)`. !!! The following references need to be updated: !!! See also Wikipedia :cite:p:`WikipediaDis08`, MathWorld :cite:p:`WolframDis08`, BoostMath :cite:p:`BoostDis08`, :cite:t:`Ehrhardt2018` (3.9.2). |cr| .. _Ctx_KolmogorovSmirnov_Pdf: .. method:: Ctx.kolmogorov_smirnov_pdf(x, a=0, b=1) where ``Ctx`` is ``Math53`` or ``CtxBoost``. Returns the pdf of the limiting form of the Kolmogorov distribution, for `x \in [0, \infty)`. 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 ("BetaPdf(x, a, b): ", BetaPdf(x, a, b)) >>> print ("dist_beta(a, b).pdf(x): ", dist_beta(a, b).pdf(x)) 6.3563523462564525615615615614561356E+00 |cr| .. _Ctx_KolmogorovSmirnov_Cdf: .. method:: Ctx.kolmogorov_smirnov_cdf(x, a=0, b=1) where ``Ctx`` is ``Math53`` or ``CtxBoost``. Returns `\displaystyle \mathrm{KF}(x) = 1-2\sum_{k=1}^{\infty}(-1)^k e^{-2k^2x^2}`, the CDF of the limiting form of the Kolmogorov distribution, for `x \in [0, \infty)`. See also :cite:t:`Ehrhardt2018` (3.9.14). 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 ("BetaCdf(x, a, b): ", BetaCdf(x, a, b)) >>> print ("dist_beta(a, b).cdf(x): ", dist_beta(a, b).cdf(x)) 6.3563523462564525615615615614561356E+00 |cr| .. _Ctx_KolmogorovSmirnov_Qtf: .. method:: Ctx.kolmogorov_smirnov_qtf(q, a=0, b=1) where ``Ctx`` is ``Math53`` or ``CtxBoost``. Returns `\mathrm{KF}^{-1}(x)`, the functional inverse of the CDF of the Kolmogorov distribution (limiting form), ie `\mathrm{KF}(\mathrm{KF}^{-1}(x)) = x`. See also :cite:t:`Ehrhardt2018` (3.9.14). 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 ("BetaQtf(q, a, b): ", BetaQtf(q, a, b)) >>> print ("dist_beta(a, b).qtf(q): ", dist_beta(a, b).qtf(q)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: CtxBoost.dist_kolmogorov_smirnov(a=0, b=1) Returns an ``dist_kolmogorov_smirnov`` 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_beta = Ctx.dist_beta(a, b) >>> print ("Dist.qtf(q=0.5): ", Dist.qtf(q=0.5)) 6.3563523462564525615615615614561356E+00 .. method:: dist_kolmogorov_smirnov.pdf(x) Returns `\text{pdf}(x)`, the value of the probability density function of the Kolmogorov distribution (limiting form). See :ref:`Ctx.kolmogorov_smirnov_pdf ` for formulas and examples. .. method:: dist_kolmogorov_smirnov.cdf(x) Returns `\text{cdf}(x)`, the value of the cumulative distribution function of the Kolmogorov distribution (limiting form). See :ref:`Ctx.kolmogorov_smirnov_cdf ` for formulas and examples. .. method:: dist_kolmogorov_smirnov.qtf(q) Returns `\text{qtf}(q)`, the value of the quantile function of the Kolmogorov distribution (limiting form). See :ref:`Ctx.kolmogorov_smirnov_qtf ` for formulas and examples. |cr| .. method:: dist_kolmogorov_smirnov.sf(x) Returns `\text{sf}(x)`, the value of the survival function (:ref:`Sf `) of the Kolmogorov distribution (limiting form). .. code-block:: pycon >>> # continued from above >>> print ("Dist.sf(x=0.5): ", Dist.qtf(x=0.5)) 6.3563523462564525615615615614561356E+00 .. method:: dist_kolmogorov_smirnov.isf(q) Returns `\text{isf}(q)`, the value of the inverse survival function (:ref:`Isf `) of the Kolmogorov distribution (limiting form). .. code-block:: pycon >>> # continued from above >>> print ("Dist.isf(x=0.5): ", Dist.isf(x=0.5)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_kolmogorov_smirnov.hf(x) Returns `\text{hazard}(x)`, the value of the hazard function (:ref:`Hf `) of the Kolmogorov distribution (limiting form). .. 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_kolmogorov_smirnov.chf(x) Returns `\text{chf}(x)`, the value of the cumulative hazard function (:ref:`Chf `) of the Kolmogorov distribution (limiting form). .. 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_kolmogorov_smirnov.mode() Returns the :ref:`mode ` of the Kolmogorov distribution (limiting form). 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_kolmogorov_smirnov.median() Returns the :ref:`median ` of the Kolmogorov distribution (limiting form). 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_kolmogorov_smirnov.mean() Returns the :ref:`mean ` (expected value) of the Kolmogorov distribution (limiting form). 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_kolmogorov_smirnov.variance() Returns the :ref:`variance ` of the Kolmogorov distribution (limiting form). 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_kolmogorov_smirnov.stdev() Returns the :ref:`standard deviation ` of the Kolmogorov distribution (limiting form). 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_kolmogorov_smirnov.skewness() Returns the :ref:`skewness ` of the Kolmogorov distribution (limiting form). 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_kolmogorov_smirnov.kurtosis() Returns the 'proper' :ref:`kurtosis ` (normalized fourth moment) of the Kolmogorov distribution (limiting form). 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_kolmogorov_smirnov.kurtosis_excess() Returns the :ref:`kurtosis excess ` of the Kolmogorov distribution (limiting form). 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_kolmogorov_smirnov.support_lower_endpoint() Returns the :ref:`support ` of the Kolmogorov distribution (limiting form) 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_kolmogorov_smirnov.support_upper_endpoint() Returns the :ref:`support ` of the Kolmogorov distribution (limiting form) 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_kolmogorov_smirnov.range_lower_endpoint() Returns the valid :ref:`range ` of the Kolmogorov distribution (limiting form) 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_kolmogorov_smirnov.range_upper_endpoint() Returns the valid :ref:`range ` of the Kolmogorov distribution (limiting form) 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