.. |newpage| raw:: latex \newpage .. |cr| raw:: latex \hspace{0.0mm} |newpage| .. _rst_dist_wilks_lambda_glm: Noncentral Distribution of Wilks' `\Lambda`: MANOVA ------------------------------------------------------------------------------- .. py:class:: ctx.dist_wilks_lambda_glm(p, m, n, omega) where ``ctx`` is ``fpm``, ``mpm``, ``ipm``, ``dec``, ``gmp`` or ``apm``. The noncentral Wilks’ `\Lambda` distribution under the GLM alternative, with `p \ge 1` predictor variables, error degress of freedom `m \ge 1` and `n \ge 1`, noncentrality parameter `\Omega` with diagonal entries `\omega_{jj} \in (0,\infty)` is a continuous probability distribution with the support interval `(0,1)`. See also :cite:t:`Anderson2003`, :cite:t:`Muirhead1982`, :cite:t:`Butler2007`, :cite:t:`Butler2002a`, :cite:t:`Butler2005`, :cite:t:`Fujikoshi1970`, :cite:t:`Fujikoshi1973`, :cite:t:`Fujikoshi1974`, :cite:t:`Lee1971a`, :cite:t:`Lee1971b`, :cite:t:`Walster1980`, :cite:t:`CharfunDis1003`. |cr| .. method:: dist_wilks_lambda_glm.pdf(x) Returns `\text{pdf}_X(x)`, the probability density function (pdf) of a random variable `X`, following the noncentral distribution of Wilks' Lambda (GLM): The pdf is computed by numerical inversion of the characteristic function or cumulant generating function. .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; x = 3; >>> print ("pdf: ", wilks_lambda_glm(mu, sigma).pdf(x)) 6.3563523462564525615615615614561356E-20 |cr| .. method:: dist_wilks_lambda_glm.cdf(x) Returns `\text{cdf}_X(x)`, the cumulative distribution function (cdf) of a random variable `X`, following the noncentral distribution of Wilks' Lambda (GLM): The cdf is computed by numerical inversion of the characteristic function or cumulant generating function. .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; x = 3; >>> print ("cdf: ", wilks_lambda_glm(mu, sigma).pdf(x)) 6.3563523462564525615615615614561356E-20 |cr| .. method:: dist_wilks_lambda_glm.sf(x) Returns `\text{sf}_X(x)`, the survival function (sf) of a random variable `X`, following the noncentral distribution of Wilks' Lambda (GLM): .. math:: \text{sf}_X(x) = 1 - \text{cdf}_X(x) = \int_{x}^{\infty} \text{pdf}_X(x) \mathrm{d} t. .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; x = 3; >>> print (" sf: ", wilks_lambda_glm(mu, sigma).pdf(x)) sf: 6.3563523462564525615615615614561356E-20 |cr| .. method:: dist_wilks_lambda_glm.qtf(q) Returns `\text{qtf}_X(x)`, the quantile function (qtf) of a random variable `X`, following the noncentral distribution of Wilks' Lambda (GLM): There is no known closed form for the quantile function `\text{cdf}^{-1}_X(q)`: It is computed with Newton iterations where the starting values are from a central chi-square approximation. .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; q = 0.3; >>> print ("qtf: ", wilks_lambda_glm(mu, sigma).qtf(q)) qtf: 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_wilks_lambda_glm.isf(q) Returns `\text{isf}_X(q)`, the inverse survival function (isf) of a random variable `X`, following the noncentral distribution of Wilks' Lambda (GLM): .. math:: \text{isf}_X(q) = \text{qtf}_X(1-q) = \text{cdf}^{-1}_X(1-q). .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; q = 0.3; >>> print ("isf: ", wilks_lambda_glm(mu, sigma).isf(q)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_wilks_lambda_glm.c_x(t) Returns `C_X(t)`, the characteristic function of a random variable `X`, following the noncentral distribution of Wilks' Lambda (GLM): .. math:: C_X(t) = \frac{\Gamma_p(n/2 -it)\Gamma_p((n + m)/2)}{\Gamma_p(n/2)\Gamma_p((n + m)/2 -it)} {}_1F_1\left( -it ;\frac{n + m}{2}+ -it ; -\frac{1}{2}\Omega\right). .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; >>> print ("c_x: ", wilks_lambda_glm(mu, sigma).c_x(t)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_wilks_lambda_glm.m_x(t) Returns `M_X(t)`, the moment generating function of a random variable `X`, following the noncentral distribution of Wilks' Lambda (GLM): .. math:: M_X(t) = \left[ \frac{\Gamma_p(n/2 + s)\Gamma_p((n + m)/2)}{\Gamma_p(n/2)\Gamma_p((n + m)/2 + s)} {}_1F_1 \left(s,\tfrac{1}{2}(n+m)+s,-\tfrac{1}{2}\Omega\right) \right]. .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; >>> print ("c_x: ", wilks_lambda_glm(mu, sigma).c_x(t)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_wilks_lambda_glm.k_x(s, k = 0) Returns `K_X(s)`, the cumulant generating function, and its `j^{\text{th}}` derivatives, `K_X^{(j)}(s), j = 1 \ldots k`, of a random variable `X`, following the noncentral distribution of Wilks' Lambda (GLM): .. math:: K_X(s) = \log \left[ \frac{\Gamma_p(n/2 + s)\Gamma_p((n + m)/2)}{\Gamma_p(n/2)\Gamma_p((n + m)/2 + s)} {}_1F_1 \left(s,\tfrac{1}{2}(n+m)+s,-\tfrac{1}{2}\Omega\right) \right]. .. math:: :nowrap: \begin{eqnarray} K'(s) & = & \sum_{i=1}^p \left[\psi \left( \tfrac{1}{2}n+s - \tfrac{1}{2}(i-1)\right) - \psi \left( \tfrac{1}{2}(n+m)+s - \tfrac{1}{2}(i-1)\right) \right] \nonumber \\ & + & \frac{\partial}{\partial s} \log \left[ {}_1F_1 \left(s,\tfrac{1}{2}(n+m)+s,-\tfrac{1}{2}\Omega\right) \right] \nonumber , \end{eqnarray} where `\Gamma_p(\cdot)` is the multivariate gamma function , `\psi(\cdot)` is the digamma function , and `{}_1F_1(\cdot,\cdot,X)` is the confluent hypergeometric function of matrix argument . .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; t = 0.3; k = 6; >>> print ("c_x: ", wilks_lambda_glm(mu, sigma).k_x(t, k)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_wilks_lambda_glm.moments(k) Returns the first `j` raw moments, `\mu_j, j = 1 \ldots k`, of a random variable `X`, following the noncentral distribution of Wilks' Lambda (GLM): .. math:: \mu'_X(r) = \frac{\Gamma_p(n/2 + s)\Gamma_p((n + m)/2)}{\Gamma_p(n/2)\Gamma_p((n + m)/2 + s)} {}_1F_1\left(s ;\frac{n + m}{2}+ s ; -\frac{1}{2}\Omega\right). .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; k = 6; >>> print ("saddlepoint: ", wilks_lambda_glm(mu, sigma).moments(k)) 6.3563523462564525615615615614561356E+00 |cr| .. method:: dist_wilks_lambda_glm.cumulants(k) Returns the first `j` cumulants, `\kappa_j, j = 1 \ldots k`, of a random variable `X`, following the noncentral distribution of Wilks' Lambda (GLM) The cumulants are calculated from the moments. .. code-block:: python >>> from mpfunlab import * >>> mp.dps = 30 >>> mu = 0; sigma = 1; k = 6; >>> print ("saddlepoint: ", wilks_lambda_glm(mu, sigma).cumulants(k)) 6.3563523462564525615615615614561356E+00 **Approximations** .. method:: ctx.wilks_lambda_glm_gp(x, p, m, n, Omega, cdf=True, method='default') where ``ctx`` is ``dec``, ``mpm``, ``fpm``, ``gmp`` or ``apm``. Calculates the pdf, cdf and sf from the characteristic function using the procedure of Gil-Pelaez (see :ref:`gil_pelaez_pdf() ` and :ref:`gil_pelaez_cdf() `). This uses `U = \log 2 W`. .. method:: ctx.wilks_lambda_glm_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. This uses `2\log W` of non-central Wilks `W` .. method:: ctx.wilks_lambda_glm_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. This uses `2\log W` of non-central Wilks `W`