Approximations based on the chi-squared distribution#
Non-Central chi-squared : cdf, sf (Patnaik)#
- ctx.chi2_nc_mu2_cdf(x)#
where
ctxisfpm,mpm,ipm,dec,gmporapm.Returns an approximation to the cdf of the noncentral chi-squared distribution by matching the first 2 moments.
Patnaik [487] gives the following 2-moment approximation based on the central \(\chi^2\)-distribution:
\[F_{\chi^2}\left(n, x; \lambda\right) \thickapprox F_{\chi^2}\left(n_1, x_1;\right), \quad \text{where } n_1= \frac{(n+\lambda)^2}{n+2\lambda} , \quad x_1= \frac{x(n+\lambda)}{n+2\lambda}\]where \(F_{\chi^2}(n, \cdot)\) is the cdf of the (central) \(\chi^2\) distribution (see section ref{sec:ChiSquareDistribution_cdf}).
An example (CDF):
>>> from mpfunlab import fpm, mpm >>> mpm.dps = 40; x = 12; nu = 10; nc = 30 >>> dx = dec.chi2_nc_mu2_cdf(x, nu, nc); mx = mpm.chi2_nc_mu2_cdf(x, nu, nc) >>> ix = ipm.chi2_nc_mu2_cdf(x, nu, nc); fx = fpm.chi2_nc_mu2_cdf(x, nu, nc) >>> gx = gmp.chi2_nc_mu2_cdf(x, nu, nc); ax = apm.chi2_nc_mu2_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%)
Non-Central chi-squared: qtf, isf (Patnaik)#
- ctx.chi2_nc_mu2_qtf(x)#
where
ctxisfpm,mpm,ipm,dec,gmporapm.Returns an approximation to the qtf of the noncentral chi-squared distribution.
The noncentral quantile is approximated as in Patnaik [487]:
\[\chi^2_{n,\lambda,\alpha} \thickapprox (1+b) \chi^2_{n_1,\alpha} , \quad \text{where } n_1= \frac{(n+\lambda)^2}{n+2\lambda} , \quad b = \frac{\lambda}{n+\lambda}\]An example (CDF):
>>> from mpfunlab import fpm, mpm >>> mpm.dps = 40; x = 12; nu = 10; nc = 30 >>> dx = dec.chi2_nc_mu2_qtf(x, nu, nc); mx = mpm.chi2_nc_mu2_qtf(x, nu, nc) >>> ix = ipm.chi2_nc_mu2_qtf(x, nu, nc); fx = fpm.chi2_nc_mu2_qtf(x, nu, nc) >>> gx = gmp.chi2_nc_mu2_qtf(x, nu, nc); ax = apm.chi2_nc_mu2_qtf(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%)
Non-Central chi-squared: confidence limit for \(\lambda\) (Winterbottom)#
- ctx.chi2_nc_wb_cl(x)#
where
ctxisfpm,mpm,ipm,dec,gmporapm.Returns an approximation to the confidence interval (CI) for the noncentrality parameter \(\lambda\) of noncentral chi-squared distribution, as decribed in Winterbottom [1606].
[Winterbottom1979` gives the following formula to determine the paramter \(\lambda\) of a noncentral \(\chi^2\) distribution with \(n\) degrees of freedom, so that \(F_{\chi^2}\left(n, x; 0\right)=1-\alpha\) and \(F_{\chi^2}\left(n, x; \lambda\right)=1-\beta\). Let c be the \(1-\alpha\) percentage point of a \(\chi^2\)-distribution with \(n\) degrees of freedom, let \(x\) be the \(1-\beta\) percentage point of a \(N(0,1)\) distribution, and \(T=(c-n)/n\), \(Y=2T+1\). Then
\begin{eqnarray} \lambda & \thickapprox & nT + \sqrt{2nY}x + \frac{2((3T+2)x^2+3T+1))}{3Y} - \frac{(6T+5)x^3-(36T^2+42T+17)x}{18\sqrt{nY^5/2}} \\ && +\: \frac{(324T^2+594T+276)x^4}{405nY^4} - \frac{(1080T^3+2484T^2+976)x^2}{405nY^4} \nonumber \\ && +\: \frac{1080T^3+1512T^2+612T+148}{405nY^4} - \frac{(10368T^3+30780T^2+30564T+10143)x^5}{9720\sqrt{n^3Y^{11}/2}} \nonumber \\ && +\: \frac{(25920T^4+98928T^3+163080T^2+137544T+47188)x^3}{9720\sqrt{n^3Y^{11}/2}}\nonumber \\ && +\: \frac{(45360T^4+106704T^3+80460T^2+31092T+13489)x}{9720\sqrt{n^3Y^{11}/2}}\nonumber \end{eqnarray}An example (CDF):
>>> from mpfunlab import fpm, mpm >>> mpm.dps = 40; x = 12; nu = 10; nc = 30 >>> dx = dec.chi2_nc_wb_cl(x, nu, nc); mx = mpm.chi2_nc_wb_cl(x, nu, nc) >>> ix = ipm.chi2_nc_wb_cl(x, nu, nc); fx = fpm.chi2_nc_wb_cl(x, nu, nc) >>> gx = gmp.chi2_nc_wb_cl(x, nu, nc); ax = apm.chi2_nc_wb_cl(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%)
Roy’s largest root \(\theta\): pdf (Chiani)#
- ctx.roy_chiani_pdf(x)#
where
ctxisfpm,mpm,ipm,dec,gmporapm.Returns an approximation to the pdf of the distribution of Roy’s largest root \(\theta\).
See Chiani [184], Chiani [185] proposes the following algorithm:
\[F_{\Theta_1}(\theta_1) \approx P \left(k, \frac{\log(\theta_1/(1-\theta_1))-\mu + \sigma \alpha}{\delta} \right)\]and for its inverse, useful for evaluating the percentiles,
\[F_{\Theta_1}^{-1}(\theta_1) \approx \frac{\exp(\sigma(\delta P^{-1}(k,y)-\alpha))+\mu}{1+\exp(\sigma(\delta P^{-1}(k,y)-\alpha))+\mu},\]where
\[\mu = 2 \log \tan \left(\frac{\gamma + \phi}{2}\right), \quad \sigma^3 = \frac{16}{(m+n+1)^2} \frac{1}{\sin^2(\gamma + \phi) \sin \gamma \sin \phi}\]\[\gamma = \arccos \left(\frac{m+n-2p}{m+n-1}\right), \quad \phi = \arccos \left(\frac{m-n}{m+n-1}\right),\]and the constants \(k = 46.446\), \(\delta = 0.186054\), \(\alpha = 9.84801\) have been chosen to match the moments of the approximation to that of the Tracy-Widom.
An example (CDF):
>>> from mpfunlab import fpm, mpm >>> mpm.dps = 40; x = 12; nu = 10; nc = 30 >>> dx = dec.roy_chiani_pdf(x, nu, nc); mx = mpm.roy_chiani_pdf(x, nu, nc) >>> ix = ipm.roy_chiani_pdf(x, nu, nc); fx = fpm.roy_chiani_pdf(x, nu, nc) >>> gx = gmp.roy_chiani_pdf(x, nu, nc); ax = apm.roy_chiani_pdf(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%)
Roy’s largest root: cdf and sf (Chiani)#
- ctx.roy_chiani_cdf(x)#
where
ctxisfpm,mpm,ipm,dec,gmporapm.Returns an approximation to the cdf of the distribution of Roy’s largest root \(\theta\).
See Chiani [184], Chiani [185] proposes the following algorithm:
\[F_{\Theta_1}(\theta_1) \approx P \left(k, \frac{\log(\theta_1/(1-\theta_1))-\mu + \sigma \alpha}{\delta} \right)\]and for its inverse, useful for evaluating the percentiles,
\[F_{\Theta_1}^{-1}(\theta_1) \approx \frac{\exp(\sigma(\delta P^{-1}(k,y)-\alpha))+\mu}{1+\exp(\sigma(\delta P^{-1}(k,y)-\alpha))+\mu},\]where
\[\mu = 2 \log \tan \left(\frac{\gamma + \phi}{2}\right), \quad \sigma^3 = \frac{16}{(m+n+1)^2} \frac{1}{\sin^2(\gamma + \phi) \sin \gamma \sin \phi}\]\[\gamma = \arccos \left(\frac{m+n-2p}{m+n-1}\right), \quad \phi = \arccos \left(\frac{m-n}{m+n-1}\right),\]and the constants \(k = 46.446\), \(\delta = 0.186054\), \(\alpha = 9.84801\) have been chosen to match the moments of the approximation to that of the Tracy-Widom.
An example (CDF):
>>> from mpfunlab import fpm, mpm >>> mpm.dps = 40; x = 12; nu = 10; nc = 30 >>> dx = dec.roy_chiani_cdf(x, nu, nc); mx = mpm.roy_chiani_cdf(x, nu, nc) >>> ix = ipm.roy_chiani_cdf(x, nu, nc); fx = fpm.roy_chiani_cdf(x, nu, nc) >>> gx = gmp.roy_chiani_cdf(x, nu, nc); ax = apm.roy_chiani_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%)
Roy’s largest root: qtf and isf (Chiani)#
- ctx.roy_chiani_qtf(x)#
where
ctxisfpm,mpm,ipm,dec,gmporapm.Returns an approximation to the qtf of the distribution of Roy’s largest root \(\theta\).
See Chiani [184], Chiani [185] proposes the following algorithm:
\[F_{\Theta_1}(\theta_1) \approx P \left(k, \frac{\log(\theta_1/(1-\theta_1))-\mu + \sigma \alpha}{\delta} \right)\]and for its inverse, useful for evaluating the percentiles,
\[F_{\Theta_1}^{-1}(\theta_1) \approx \frac{\exp(\sigma(\delta P^{-1}(k,y)-\alpha))+\mu}{1+\exp(\sigma(\delta P^{-1}(k,y)-\alpha))+\mu},\]where
\[\mu = 2 \log \tan \left(\frac{\gamma + \phi}{2}\right), \quad \sigma^3 = \frac{16}{(m+n+1)^2} \frac{1}{\sin^2(\gamma + \phi) \sin \gamma \sin \phi}\]\[\gamma = \arccos \left(\frac{m+n-2p}{m+n-1}\right), \quad \phi = \arccos \left(\frac{m-n}{m+n-1}\right),\]and the constants \(k = 46.446\), \(\delta = 0.186054\), \(\alpha = 9.84801\) have been chosen to match the moments of the approximation to that of the Tracy-Widom.
An example (CDF):
>>> from mpfunlab import fpm, mpm >>> mpm.dps = 40; x = 12; nu = 10; nc = 30 >>> dx = dec.roy_chiani_qtf(x, nu, nc); mx = mpm.roy_chiani_qtf(x, nu, nc) >>> ix = ipm.roy_chiani_qtf(x, nu, nc); fx = fpm.roy_chiani_qtf(x, nu, nc) >>> gx = gmp.roy_chiani_qtf(x, nu, nc); ax = apm.roy_chiani_qtf(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%)