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|newpage| Approximations based on the noncentral chi-squared distribution =============================================================================== .. _rst_mpm_wilks_lambda_glm_chi2_cdf: Non-central Wilks' Lambda (GLM): cdf and sf (Fujikoshi) ---------------------------------------------------------------------------------------------------- .. method:: ctx.wilks_lambda_glm_chi2_cdf(x, p, q, n, lambda) where ``ctx`` is ``fpm``, ``mpm``, ``ipm``, ``dec``, ``gmp`` or ``apm``. Returns an approximation to the cdf of the noncentral Wilks' Lambda (GLM) distribution. :cite:t:`Fujikoshi1973` proposes the following noncentral chi-square approximation to the cdf and sf: .. math:: :nowrap: \begin{eqnarray} F_{\Lambda,GLM}(p,q,n;z;\Theta) & = & F_{\chi^2}\left(pq, x; \theta_1\right) + \frac{1}{4n} \sum_{k=1}^3{a_k } F_{\chi^2}\left(pq+2k, x; \theta_1\right) \\ & + & \frac{1}{96n^2} \sum_{k=0}^6{b_k } F_{\chi^2}\left(pq+2k, x; \theta_1\right) \nonumber + \frac{1}{96n^3} \sum_{k=1}^9{c_k } F_{\chi^2}\left(pq+2k, x; \theta_1\right) + O(n^{-4}) \nonumber \end{eqnarray} .. math:: \Theta = \Lambda^{-1} M M', \quad \theta_j = \text{tr } \Theta^j, .. math:: m=n+(pq-1)/2, \quad x= m \log(z), \quad s=(p+q+1)/4, \quad r=pq (p^2+q^2-5)/48, .. math:: :nowrap: \begin{eqnarray} a_1 & = & 2 s \theta_1 \\ a_2 & = & 2 s \theta_1 - \theta_2 \nonumber \\ a_3 & = & \theta_2 \nonumber \\ \nonumber \\ b_0 & = & r \\ b_1 & = & 0 \nonumber \\ b_2 & = & r 4s^2 \theta_1 + 2s^2 \theta_1^2 + 2s \theta_2 \nonumber \\ b_3 & = & 4 s^2 \theta_1 - (1+4 s^2) \theta_1^2 - (1+8s) \theta_2 + 2 s \theta_1 \theta_2 + (4/3) \theta_3 \nonumber \\ b_4 & = & (1 + 2s^2) \theta_1^2 + (1+6s)\theta_2 - 4s\theta_1\theta_2- 4\theta_3 + \theta_2^2/2 \nonumber \\ b_5 & = & 2s\theta_1\theta_2 + (8/3)\theta_3 - \theta_2^2 \nonumber \\ b_6 & = & \theta_2^2/2 \nonumber \\ \nonumber \\ c_1 & = & 2 r s \theta_1 \\ c_2 & = & r (2 s \theta_1 \theta_2) \nonumber \\ c_3 & = & 2s(r+4s^2)\theta_1 + 2s(1+4 s^2)\theta_1^2 + (-r+2s+12s^2)\theta_2 - \tfrac{4}{3}s^3 \theta_1^3 -4s^2\theta_1\theta_2 -\tfrac{8}{3}s\theta_3 \nonumber \\ c_4 & = & 2 s (r+4 s^2) \theta_1 - (1+10s+16s^3)\theta_1^2 - (3+r+10s+36s^2) \theta_2 + 2s(1+2s^2)\theta_1^3 \nonumber \\ & + & 2(2+s+12s^2)\theta_1\theta_2 + 4(1+6s)\theta_3 - 2s^2\theta_1^2\theta_2 - 2s\theta_2^2 - \tfrac{8}{3}s\theta_1\theta_3 - 2\theta_4 \nonumber \\ c_5 & = & (1+8 s+8 s^3)\theta_1^2 + (3+r+8s+24s^2)\theta_2 - 4s(1+s^2)\theta_1^3 - 4(3+s+9s^2)\theta_1\theta_2 - 12(1+4s)\theta_3 \nonumber \\ & + & (1+6s^2)\theta_1^2 \theta_2 + (1+10s)\theta_2^2 + \tfrac{32}{3}s\theta_1\theta_3 + 12\theta_4 - \tfrac{4}{3}\theta_2 \theta_3 - s\theta_1\theta_2^2 \nonumber \\ c_6 & = & s (2+\tfrac{4}{3}s^2) \theta_1^3+2 (4+s+8 s^2) \theta_1 \theta_2+8 (1+\tfrac{10}{3}s) \theta_3 - 2 (1+3 s^2) \theta_1^2 \theta_2 - 2 (1+7 s) \theta_2^2 \nonumber \\ & - & \tfrac{40}{3}s \theta_1 \theta_3 - 20\theta_4 + \tfrac{16}{3}\theta_2\theta_3 + 3s\theta_1\theta_2^2 - \tfrac{1}{6} \theta_2^3 \nonumber \\ c_7 & = & (1+2 s^2)\theta_1^2\theta_2 + (1+6 s)\theta_2^2 + \tfrac{16}{3}s\theta_1\theta_3 + 10\theta_4 - \tfrac{20}{3}\theta_2\theta_3 - 3s\theta_1\theta_2^2 + \tfrac{1}{2}\theta_2^3 \nonumber \\ c_8 & = & \tfrac{8}{3}\theta_2\theta_3 + s\theta_1\theta_2^2 - \tfrac{1}{2}\theta_2^3 \nonumber \\ c_9 & = & \tfrac{1}{6}\theta_2^3 \nonumber \end{eqnarray} An example (CDF): .. code-block:: pycon >>> from mpfunlab import fpm, mpm >>> mpm.dps = 40; x = 12; nu = 10; nc = 30 >>> dx = dec.wilks_lambda_glm_chi2_cdf(x, nu, nc); mx = mpm.wilks_lambda_glm_chi2_cdf(x, nu, nc) >>> ix = ipm.wilks_lambda_glm_chi2_cdf(x, nu, nc); fx = fpm.wilks_lambda_glm_chi2_cdf(x, nu, nc) >>> gx = gmp.wilks_lambda_glm_chi2_cdf(x, nu, nc); ax = apm.wilks_lambda_glm_chi2_cdf(x, nu, nc) >>> mpm.show([dx, mx, ix, fx, gx, ax]) dec: 3.113877423416836055714616090074444149943E-1 mpm: 3.113877423416836055714616090074444149943e-1 ipm: 3.113877423416836055714616090074444149944e-1 (4.147e-38%) fpm: 3.11387742341684E-01 gmp: 3.113877423416836055714616090074444149943E-01 ipm: 3.113877423416836055714616090074444149943e-1 (4.055e-38%) |newpage| :cite:t:`Fujikoshi1973` Pillai: coeff p. 93 (5.10), GLM p. 96 (5.2), CORR p. 113/114 (8.1) Hotelling: coeff p. 105 (6.18), GLM p. 105 (6.19), CORR p. 119 (9.9) .. _rst_mpm_wilks_lambda_ind_chi2_cdf: Non-central Wilks' Lambda (independence): cdf and sf (Lee) ---------------------------------------------------------------------------------------------------- .. method:: ctx.wilks_lambda_ind_chi2_cdf(x, p, q, n, lambda) where ``ctx`` is ``fpm``, ``mpm``, ``ipm``, ``dec``, ``gmp`` or ``apm``. Returns an approximation to the cdf of the noncentral Wilks' Lambda (independence) distribution. :cite:t:`Lee1971b` proposes the following noncentral chi-square approximation: .. math:: \text{cdf}_X(x) = = F_{\chi^2}\left(pq, x; s1\right) + \frac{1}{4n} \sum_{k=0}^3{a_k } F_{\chi^2}\left(pq+2k, x; s1\right) + \frac{1}{96n^2} \sum_{k=0}^6{b_k } F_{\chi^2}\left(pq+2k, x; s1\right) + O(n^{-3}), `\Omega = \Lambda^{-1} M M'`, `\theta_j = \text{tr } \Omega^j`, `m=n+(pq-1)/2`, `x= m \log(z)`, `f = p q`, `s=(p+q+1)/4`, `r=f (p^2+q^2-5)/48`, `a_0=-q \theta_1+\theta_2`, `a_1=(2 s+q) \theta_1 - 2 \theta_2`, `a_2=-2 s \theta_1+2 \theta_2`, `a_3=-\theta_2`, `b_0= -r - q l \theta_1+(q+l) \theta_2 + \tfrac{1}{2} q^2 \theta_1^2 - \tfrac{4}{3} \theta_3 - q \theta_1 \theta_2 + \tfrac{1}{2} \theta_2^2, \\` `b_1=q^2 \theta_1 - 4 q \theta_2 - q (q+2s) \theta_1^2 + 4 \theta_3+(3q+2s) \theta_1 \theta_2 - 2 \theta_2^2, \\` `b_2= r - 2s (q+2s) \theta_1 + (2p+6q+3) \theta_2 + (\tfrac{1}{2} l^2+6qs+1) \theta_1^2 - 8 \theta_3 -(4q+6s) \theta_1 \theta_2 + 4 \theta_2^2, \\` `b_3=4s^2 \theta_1 -(3p+5q+5) \theta_2 -(4s^2+2qs+2) \theta_1^2 + \tfrac{32}{3} \theta_3 +(3q+8s) \theta_1 \theta_2 - 5\theta_2^2, \\` `b_4=(6s+1) \theta_2 + (2s^2+1) \theta_1^2 - 8\theta_3 -(q+6s)\theta_1\theta_2 + 4 \theta_2^2, \\` `b_5 = \tfrac{8}{3} \theta_3 + 2s\theta_1\theta_2 - 2\theta_2^2, \\` `b_6=\tfrac{1}{2} \theta_2^2`. An example (CDF): .. code-block:: pycon >>> from mpfunlab import fpm, mpm >>> mpm.dps = 40; x = 12; nu = 10; nc = 30 >>> dx = dec.wilks_lambda_ind_chi2_cdf(x, nu, nc); mx = mpm.wilks_lambda_ind_chi2_cdf(x, nu, nc) >>> ix = ipm.wilks_lambda_ind_chi2_cdf(x, nu, nc); fx = fpm.wilks_lambda_ind_chi2_cdf(x, nu, nc) >>> gx = gmp.wilks_lambda_ind_chi2_cdf(x, nu, nc); ax = apm.wilks_lambda_ind_chi2_cdf(x, nu, nc) >>> mpm.show([dx, mx, ix, fx, gx, ax]) dec: 3.113877423416836055714616090074444149943E-1 mpm: 3.113877423416836055714616090074444149943e-1 ipm: 3.113877423416836055714616090074444149944e-1 (4.147e-38%) fpm: 3.11387742341684E-01 gmp: 3.113877423416836055714616090074444149943E-01 ipm: 3.113877423416836055714616090074444149943e-1 (4.055e-38%) |newpage| .. _rst_mpm_pillai_v_glm_chi2_cdf: Non-central Pillai's V (GLM): cdf and sf Fujikoshi ---------------------------------------------------------------------------------------------------- .. method:: ctx.pillai_v_glm_chi2_cdf(x, p, q, n, lambda) where ``ctx`` is ``fpm``, ``mpm``, ``ipm``, ``dec``, ``gmp`` or ``apm``. Returns an approximation to the cdf of the noncentral Pillai's V (GLM) distribution. :cite:t:`Fujikoshi1974` proposes the following noncentral chi-square approximation to the cdf and sf: .. math:: \text{cdf}_X(x) = = F_{\chi^2}\left(pq, x; s1\right) + \frac{1}{4n} \sum_{k=0}^3{a_k } F_{\chi^2}\left(pq+2k, x; s1\right) + \frac{1}{96n^2} \sum_{k=0}^6{b_k } F_{\chi^2}\left(pq+2k, x; s1\right) + O(n^{-3}), `l_0 = (3 f - 8) g^2 + 4 g + 4 (f + 2)` `l_1 = -12 f g^2` `l_2 = 6 (3 f + 8) g^2` `l_3 = -4 ((3 f + 16) g^2 + 4 g + 4 (f + 2))` `l_4 = 3 ((f + 8) g^2 + 4 g + 4 (f + 2))` `a_0 = -f g` `a_1 = 2 f g` `a_2 = -f g + 4 g \lambda_1 + 4 \lambda_2` `a_3 = -4 g \lambda_1` `a_4 = -4 \lambda_2` `b_0 = f l_0` `b_1 = f l_1` `b_2 = f l_2 + 2 l_1 \lambda_1 - 24 f g \lambda_2` `b_3 = f l_3 + 4 l_2 \lambda_1 + 48 (f + 4) g \lambda_2 + 128 \lambda_3` `b_4 = f l_4 + 6 l_3 \lambda_1 + 48 (g^2 - 2) \lambda_1^2 - 96 (g + 1) \lambda_2 + 96 g \lambda_1 \lambda_2 + 48 \lambda_2^2` `b_5 = 8 (l_4 \lambda_1 - 12 (g^2 + 2) \lambda_1^2 - 6 ((f + 12) g + 4) \lambda_2 - 12 g \lambda_1 \lambda_2 - 48 \lambda_3)` `b_6 = 8 (6 (g^2 + 6) \lambda_1^2 + 3 ((f + 20) g + 12) \lambda_2 - 12 g \lambda_1 \lambda_2 - 16 \lambda_3 - 12 \lambda_2^2)` `b_7 = 96 (g \lambda_1 \lambda_2 + 4 \lambda_3)` `b_8 = 48 \lambda_2^2` An example (CDF): .. code-block:: pycon >>> from mpfunlab import fpm, mpm >>> mpm.dps = 40; x = 12; nu = 10; nc = 30 >>> dx = dec.pillai_v_glm_chi2_cdf(x, nu, nc); mx = mpm.pillai_v_glm_chi2_cdf(x, nu, nc) >>> ix = ipm.pillai_v_glm_chi2_cdf(x, nu, nc); fx = fpm.pillai_v_glm_chi2_cdf(x, nu, nc) >>> gx = gmp.pillai_v_glm_chi2_cdf(x, nu, nc); ax = apm.pillai_v_glm_chi2_cdf(x, nu, nc) >>> mpm.show([dx, mx, ix, fx, gx, ax]) dec: 3.113877423416836055714616090074444149943E-1 mpm: 3.113877423416836055714616090074444149943e-1 ipm: 3.113877423416836055714616090074444149944e-1 (4.147e-38%) fpm: 3.11387742341684E-01 gmp: 3.113877423416836055714616090074444149943E-01 ipm: 3.113877423416836055714616090074444149943e-1 (4.055e-38%) |newpage| .. _rst_mpm_pillai_v_ind_chi2_cdf: Non-central Pillai's V (independence): cdf and sf (Lee) ---------------------------------------------------------------------------------------------------- .. method:: ctx.pillai_v_ind_chi2_cdf(x, p, q, n, lambda) where ``ctx`` is ``fpm``, ``mpm``, ``ipm``, ``dec``, ``gmp`` or ``apm``. Returns an approximation to the cdf of the noncentral Pillai's V (independence) distribution. :cite:t:`Lee1971b` proposes the following noncentral chi-square approximation. See also .. math:: \text{cdf}_X(x) = = F_{\chi^2}\left(pq, x; s1\right) + \frac{1}{4n} \sum_{k=0}^3{a_k } F_{\chi^2}\left(pq+2k, x; s1\right) + \frac{1}{96n^2} \sum_{k=0}^6{b_k } F_{\chi^2}\left(pq+2k, x; s1\right) + O(n^{-3}), `a_0 = -f g - 4 \lambda_2` `a_1 = 2 f g` `a_2 = -f g + 4 g \lambda_1 + 8 \lambda_2` `a_3 = -4 g \lambda_1` `a_4 = -4 \lambda_2` `b_0 = f l_0 + 24 f g \lambda_2 - 128 \lambda_3 + 48 \lambda_2^2` `b_1 = f l_1 - 48 f g \lambda_2` `b_2 = f l_2 + 2 l_1 \lambda_1 + 96 \lambda_1^2 - 24 (q p^2 + q (q + 1) p - 4) \lambda_2 - 96 g \lambda_1 \lambda_2 - 192 \lambda_2^2` `b_3 = f l_3 + 4 l_2 \lambda_1 + 96 (q p^2 + (q^2 + q + 4) p + 4 (q + 1)) \lambda_2 + 96 g \lambda_1 \lambda_2 + 640 \lambda_3` `b_4 = f l_4 + 6 l_3 \lambda_1 + 48 (p^2 + 2 (q + 1) p + q^2 + 2 q - 3) \lambda_1^2 - 24 (q p^2 + (q^2 + q + 12) p + 4 (3 q + 5)) \lambda_2 + 192 g \lambda_1 \lambda_2 + 288 \lambda_2^2` `b_5 = 8 l_4 \lambda_1 - 96 (p^2 + 2 (q + 1) p + q^2 + 2 q + 3) \lambda_1^2 - 48 (q p^2 + (q^2 + q + 12) p + 4 (3 q + 4)) \lambda_2 - 192 g \lambda_1 \lambda_2 - 768 \lambda_3` `b_6 = 48 (p^2 + 2 (q + 1) p + q^2 + 2 q + 7) \lambda_1^2 + 24 (q p^2 + (q^2 + q + 20) p + 4 (5 q + 8)) \lambda_2 - 96 g \lambda_1 \lambda_2 - 128 \lambda_3 - 192 \lambda_2^2` `b_7 = 96 (g \lambda_1 \lambda_2 + 4 \lambda_3)` `b_8 = 48 \lambda_2^2` An example (CDF): .. code-block:: pycon >>> from mpfunlab import fpm, mpm >>> mpm.dps = 40; x = 12; nu = 10; nc = 30 >>> dx = dec.pillai_v_ind_chi2_cdf(x, nu, nc); mx = mpm.pillai_v_ind_chi2_cdf(x, nu, nc) >>> ix = ipm.pillai_v_ind_chi2_cdf(x, nu, nc); fx = fpm.pillai_v_ind_chi2_cdf(x, nu, nc) >>> gx = gmp.pillai_v_ind_chi2_cdf(x, nu, nc); ax = apm.pillai_v_ind_chi2_cdf(x, nu, nc) >>> mpm.show([dx, mx, ix, fx, gx, ax]) dec: 3.113877423416836055714616090074444149943E-1 mpm: 3.113877423416836055714616090074444149943e-1 ipm: 3.113877423416836055714616090074444149944e-1 (4.147e-38%) fpm: 3.11387742341684E-01 gmp: 3.113877423416836055714616090074444149943E-01 ipm: 3.113877423416836055714616090074444149943e-1 (4.055e-38%) |newpage| .. _rst_mpm_hotelling_t2_glm_chi2_cdf: Non-central Hotelling `T^2` (GLM): cdf and sf (Fujikoshi) ---------------------------------------------------------------------------------------------------- .. method:: ctx.hotelling_t2_glm_chi2_cdf(x) where ``ctx`` is ``fpm``, ``mpm``, ``ipm``, ``dec``, ``gmp`` or ``apm``. Returns an approximation to the cdf of the noncentral Hotelling `T^2` (GLM) distribution. :cite:t:`Fujikoshi1974` proposes the following noncentral chi-square approximation to the pdf and cdf: .. math:: \text{cdf}_X(x) = = F_{\chi^2}\left(pq, x; s1\right) + \frac{1}{4n} \sum_{k=0}^3{a_k } F_{\chi^2}\left(pq+2k, x; s1\right) + \frac{1}{96n^2} \sum_{k=0}^6{b_k } F_{\chi^2}\left(pq+2k, x; s1\right) + O(n^{-3}), `a_0 = f g` `a_1 = -2 g (f - 2 \lambda_1)` `a_2 = f g - 8 g \lambda_1 + 4 \lambda_2` `a_3 = 4 (g \lambda_1 - 2 \lambda_2)` `a_4 = 4 \lambda_2` `b_0 = f l_0` `b_1 = l_1 (f - 2 \lambda_1)` `b_2 = f l_2 + 2 (l_1 - 2 l_2) \lambda_1 + 48 g^2 \lambda_1^2 + 24 (f + 4) g \lambda_2` `b_3 = f l_3 + 2 (2 l_2 - 3 l_3) \lambda_1 - 192 (g^2 + 1) \lambda_1^2 - 96 ((f + 8) g + 2) \lambda_2 + 96 g \lambda_1 \lambda_2 + 128 \lambda_3` `b_4 = f l_4 + 2 (3 l_3 - 4 l_4) \lambda_1 + 96 (3 g^2 + 7) \lambda_1^2 + 48 (3 (f + 12) g + 14) \lambda_2 - 384 g \lambda_1 \lambda_2 - 768 \lambda_3 + 48 \lambda_2^2` `b_5 = 8 l_4 \lambda_1 - 192 (g^2 + 4) \lambda_1^2 - 96 ((f + 16) g + 8) \lambda_2 + 576 g \lambda_1 \lambda_2 + 1536 \lambda_3 - 192 \lambda_2^2` `b_6 = 48 (g^2 + 6) \lambda_1^2 + 24 ((f + 20) g + 12) \lambda_2 - 384 g \lambda_1 \lambda_2 - 1280 \lambda_3 + 288 \lambda_2^2` `b_7 = 96 g \lambda_1 \lambda_2 + 384 \lambda_3 - 192 \lambda_2^2` `b_8 = 48 \lambda_2^2` An example (CDF): .. code-block:: pycon >>> from mpfunlab import fpm, mpm >>> mpm.dps = 40; x = 12; nu = 10; nc = 30 >>> dx = dec.hotelling_t2_glm_chi2_cdf(x, nu, nc); mx = mpm.hotelling_t2_glm_chi2_cdf(x, nu, nc) >>> ix = ipm.hotelling_t2_glm_chi2_cdf(x, nu, nc); fx = fpm.hotelling_t2_glm_chi2_cdf(x, nu, nc) >>> gx = gmp.hotelling_t2_glm_chi2_cdf(x, nu, nc); ax = apm.hotelling_t2_glm_chi2_cdf(x, nu, nc) >>> mpm.show([dx, mx, ix, fx, gx, ax]) dec: 3.113877423416836055714616090074444149943E-1 mpm: 3.113877423416836055714616090074444149943e-1 ipm: 3.113877423416836055714616090074444149944e-1 (4.147e-38%) fpm: 3.11387742341684E-01 gmp: 3.113877423416836055714616090074444149943E-01 ipm: 3.113877423416836055714616090074444149943e-1 (4.055e-38%) |newpage| .. _rst_mpm_hotelling_t2_ind_chi2_cdf: Non-central Hotelling `T^2` (independence): cdf and sf (Lee) ---------------------------------------------------------------------------------------------------- .. method:: ctx.hotelling_t2_ind_chi2_cdf(x) where ``ctx`` is ``fpm``, ``mpm``, ``ipm``, ``dec``, ``gmp`` or ``apm``. Returns an approximation to the cdf of the noncentral Hotelling `T^2` (independence) distribution. :cite:t:`Lee1971b` proposes the following noncentral chi-square approximation: .. math:: \text{cdf}_X(x) = = F_{\chi^2}\left(pq, x; s1\right) + \frac{1}{4n} \sum_{k=0}^3{a_k } F_{\chi^2}\left(pq+2k, x; s1\right) + \frac{1}{96n^2} \sum_{k=0}^6{b_k } F_{\chi^2}\left(pq+2k, x; s1\right) + O(n^{-3}), `s_1 = 2\lambda_1` `s_2 = 4\lambda_2` `s_3 = 8\lambda_3` `a_0 = q p (q - p - 1) - 2 q s_1 + s_2` `a_1 = -2 q^2 p + 4 q s_1 - 2 s_2` `a_2 = q p (q + p + 1) - 2 (2 q + p + 1) s_1 + 2 s_2` `a_3 = 2 (q + p + 1) s_1 - 2 s_2` `a_4 = s_2` `b_0 = q p (3 q p^3 - 2 (3 q^2 - 3 q + 4) p^2 + 3 (q^3 - 2 q^2 + 5 q - 4) p - 8 q^2 + 12 q + 4) - 12 q^2 p (q - p - 1) s_1 - 6 q (p^2 - q p + p - 4) s_2 + 12 q^2 s_1^2 - 16 s_3 - 12 q s_1 s_2 + 3 s_22` `b_1 = -12 q^3 p^2 (q - p - 1) - 24 q^2 (p^2 - 2 q p + p - 2) s_1 + 12 q (p^2 - 2 q p + p - 8) s_2 - 48 q^2 s_1^2 + 48 s_3 + 48 q s_1 s_2 - 12 s_22` `b_2 = -6 q^2 p^4 - 12 q^2 p^3 + 18 q^2 (q^2 + 1) p^2 + 24 q^2 (2 q + 1) p + 12 q (p^3 + 2 p^2 - 7 (q^2 + 1) p - 16 q - 8) s_1 - 6 (q p^2 - (7 q^2 - q + 8) p - 40 q - 12) s_2 + 24 (q p + 4 q^2 + q + 1) s_1^2 - 12 (p + 8 q + 1) s_1 s_2 - 96 s_3 + 24 s_22` `b_3 = -(12 q^3 + 16 q) p^3 - (12 q^4 + 12 q^3 + 96 q^2 + 48 q) p^2 - (64 q^3 + 96 q^2 + 64 q) p + 12 (-q p^3 + (4 q^2 - 2 q + 4) p^2 + (7 q^3 + 4 q^2 + 31 q + 12) p + 4 (7 q^2 + 8 q + 4)) s_1 - 48 ((q^2 + 3) p + 9 q + 5) s_2 - 24 (3 q p + 5 q^2 + 3 q + 4) s_1^2 + 176 s_3 + 12 (3 p + 11 q + 3) s_1 s_2 - 36 s_22` `b_4 = 3 q^2 p^4 + (6 q^3 + 6 q^2 + 24 q) p^3 + (3 q^4 + 6 q^3 + 63 q^2 + 60 q) p^2 + (24 q^3 + 60 q^2 + 60 q) p - 12 (q p^3 + (5 q^2 + 2 q + 12) p^2 + (4 q^3 + 5 q^2 + 45 q + 32) p + 4 (6 q^2 + 11 q + 9)) s_1 + 6 (q p^2 + (7 q^2 + q + 44) p + 88 q + 76) s_2 + 12 (p^2 + 2 (4 q + 1) p + 8 q^2 + 8 q + 17) s_1^2 - 12 (4 p + 11 q + 4) s_1 s_2 - 240 s_3 + 42 s_22` `b_5 = (12 q p^3 + 24 (q^2 + q + 4) p^2 + 12 (q^3 + 2 q^2 + 21 q + 20) p + 48 (2 q^2 + 5 q + 5)) s_1 - 12 (q p^2 + (2 q^2 + q + 24) p + 32 q + 40) s_2 - 24 (p^2 + (3 q + 2) p + 2 q^2 + 3 q + 9) s_1^2 + 240 s_3 + 48 (p + 2 q + 1) s_1 s_2 - 36 s_22` `b_6 = (6 q p^2 + 6 (q^2 + q + 20) p + 120 q + 192) s_2 + (12 p^2 + 24 (q + 1) p + 12 (q^2 + 2 q + 7)) s_1^2 - 12 (3 p + 4 q + 3) s_1 s_2 - 160 s_3 + 24 s_22` `b_7 = 48 s_3 + 12 (q + p + 1) s_1 s_2 - 12 s_22` `b_8 = 3 s_22` An example (CDF): .. code-block:: pycon >>> from mpfunlab import fpm, mpm >>> mpm.dps = 40; x = 12; nu = 10; nc = 30 >>> dx = dec.hotelling_t2_ind_chi2_cdf(x, nu, nc); mx = mpm.hotelling_t2_ind_chi2_cdf(x, nu, nc) >>> ix = ipm.hotelling_t2_ind_chi2_cdf(x, nu, nc); fx = fpm.hotelling_t2_ind_chi2_cdf(x, nu, nc) >>> gx = gmp.hotelling_t2_ind_chi2_cdf(x, nu, nc); ax = apm.hotelling_t2_ind_chi2_cdf(x, nu, nc) >>> mpm.show([dx, mx, ix, fx, gx, ax]) dec: 3.113877423416836055714616090074444149943E-1 mpm: 3.113877423416836055714616090074444149943e-1 ipm: 3.113877423416836055714616090074444149944e-1 (4.147e-38%) fpm: 3.11387742341684E-01 gmp: 3.113877423416836055714616090074444149943E-01 ipm: 3.113877423416836055714616090074444149943e-1 (4.055e-38%)