.. |newpage| raw:: latex \newpage .. |cr| raw:: latex \hspace{0.0mm} |newpage| .. _rst_dist_page_l_continuous: Generalized Page `L` distribution, continuous data ------------------------------------------------------------------------------- .. py:class:: ctx.dist_page_l(k, n) where ``ctx`` is ``fpm``, ``mpm``, ``ipm``, ``dec``, ``gmp`` or ``apm``. The distribution of Page's `L` is a discrete (lattice) probability distribution with k samples of size `n_1 \ge 1, \ldots, n_k \ge 1` and the support interval `(0, n m))`. See also Wikipedia :cite:p:`WikipediaDis29`, :cite:t:`Noether1967`, :cite:t:`vandeWiel2000` . Consider `k` independent groups `X_i` of sizes `n_i, i=1 \ldots k`. The Page `L` statistic is defined as .. math:: L = ?? |cr| .. method:: dist_page_l.pmf(x) Returns `\text{pmf}_X(x)`, the probability mass function (pmf) of a random variable `X`, following a generalized Page `L` distribution. .. math:: \text{pmf}_X(x) = ?? .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; x = 3; >>> print ("pmf: ", page_l_continuous(mu, sigma).pmf(x)) 6.3563523462564525615615615614561356E-20 |cr| .. method:: dist_page_l.cdf(x) Returns `\text{cdf}_X(x)`, the cumulative distribution function (cdf) of a random variable `X`, following a generalized Page `L` distribution. .. math:: \text{cdf}_X(x) = ?? .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; x = 3; >>> print ("cdf: ", page_l_continuous(mu, sigma).pmf(x)) 6.3563523462564525615615615614561356E-20 |cr| .. method:: dist_page_l.sf(x) Returns `\text{sf}_X(x)`, the survival function (sf) of a random variable `X`, following a generalized Page `L` distribution: .. math:: \text{sf}_X(x) = 1 - \text{cdf}_X(x) .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; x = 3; >>> print (" sf: ", page_l_continuous(mu, sigma).pmf(x)) sf: 6.3563523462564525615615615614561356E-20 |cr| .. method:: dist_page_l.qtf(q) Returns `\text{qtf}_X(x)`, the quantile function (qtf) of a random variable `X`, following a generalized Page `L` distribution. There is no closed form for the qtf: It is computed with Newton iterations where the starting values are from Boost. .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; q = 0.3; >>> print ("qtf: ", page_l_continuous(mu, sigma).qtf(q)) qtf: 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_page_l.isf(q) Returns `\text{isf}_X(q)`, the inverse survival function (isf) of a random variable `X`, following a generalized Page `L` distribution: .. math:: \text{isf}_X(q) = \text{qtf}_X(1-q). .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; q = 0.3; >>> print ("isf: ", page_l_continuous(mu, sigma).isf(q)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_page_l.c_x(t) Returns `C_X(t)`, the characteristic function of a random variable `X`, following a generalized Page `L` distribution: .. math:: C_X(t) = ?? .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; >>> print ("c_x: ", page_l_continuous(mu, sigma).c_x(t)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_page_l.m_x(t) Returns `M_X(t)`, the moment generating function of a random variable `X`, following a generalized Page `L` distribution: .. math:: M_X(t) = ?? .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; >>> print ("m_x: ", page_l_continuous(mu, sigma).c_x(t)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_page_l.k_x(t, k = 0) Returns `K_X(t)`, the cumulant generating function of a random variable `X`, following a generalized Page `L` distribution: .. math:: K_X(t) = ?? .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; k = 6; >>> print ("c_x: ", page_l_continuous(mu, sigma).k_x(t, k)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_page_l.moments(k) Returns the first `j` moments, `\mu_j, j = 1 \ldots k`, of a random variable `X`, following a generalized Page `L` distribution (Wikipedia). The moments are calculated from the cumulants. .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; k = 6; >>> print ("saddlepoint: ", page_l_continuous(mu, sigma).moments(k)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_page_l.cumulants(k) Returns the first `j` cumulants, `\kappa_j, j = 1 \ldots k`, of a random variable `X`, following a generalized Page `L` distribution. The cumulants of `J_N` are given by : .. math:: \kappa_{2j} = ?? .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; k = 6; >>> print ("saddlepoint: ", page_l_continuous(mu, sigma).cumulants(k)) 6.3563523462564525615615615614561356E+00 **Approximations** .. method:: ctx.page_ecf(x, f, rho, omega) where ``ctx`` is ``ipm``, ``dec``, ``mpm``, or ``gmp``. Calculates the Edgeworth approximation to the pdf, cdf and sf. For the Page L distribution, the cumulants are calculated from the pmf .. method:: ctx.page_ecf_inv(q, f, rho, omega) where ``ctx`` is ``ipm``, ``dec``, ``mpm``, or ``gmp``. Calculates the Cornish-Fisher approximation to the qtf and isf. For the Page L distribution, the cumulants are calculated from the pmf .. method:: ctx.page_spa(x, n, results='c') where ``ctx`` is ``fpm``, ``mpm``, ``ipm``, ``dec``, ``gmp`` or ``apm``. Calculates the Luggannini-Rice saddlepoint approximation of the pdf, cdf and sf. `K_X(t)`, the cumulant generating function, and its `j^{\text{th}}` derivatives, `K_X^{(j)}(t), j = 1 \ldots k`, of a random variable `X`, following a Jonckheere-Terpsta S distribution, are calculated from the pgf. The saddlepoint `s` is determined numerically using Newton iterations, with a starting value of `s=0.1`. .. method:: ctx.page_spa_inv(x, n, results='qtf') where ``ctx`` is ``fpm``, ``mpm``, ``ipm``, ``dec``, ``gmp`` or ``apm``. Calculates the inverse Jensen saddlepoint approximation of the qtf and isf.