Pearson’s rho distribution (under \(H_0\))

class ctx.dist_pearson_rho(N)

where ctx is fpm, mpm, ipm, dec, gmp or apm.

The distribution of Pearson’s rho (under \(H_0\)) with sample size \(N \ge 3\), ie. the distribution of the sample correlation coefficient when \(\rho=0\)), is a continuous distribution with the support interval \((-1,+1)\).

See also: Wikipedia [1326], Johnson et al. [410] page 550.

See also Johnson II, page 550, for characteristic function and mgf.

dist_pearson_rho.pdf(x)

Returns \(\text{pdf}_X(x)\), the probability density function (pdf) of a random variable \(X\), following the distribution of Pearson’s rho (under \(H_0\)):

\[\text{pdf}_X(x) = {\frac {(1-r^{2})^{\frac {N-4}{2}}}{B\left({\frac {1}{2}},{\frac {N-2}{2}}\right)}},\]

where \(B(a,b)\) is the beta function.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("pdf: ", pearson_rho(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_pearson_rho.cdf(x)

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following the distribution of Pearson’s rho (under \(H_0\)):

\[\text{cdf}_X(r) = F_{\text{StudentT}}\left(t, N-2\right), \quad \text{where } t = r \sqrt{\frac{N-2}{1-r^2}},\]

and \(F_{\text{StudentT}}\left(t, N-2,\right)\) is the cdf of the t-distribution with \(N-2\) degrees of freedom.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("cdf: ", pearson_rho(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_pearson_rho.sf(x)

Returns \(\text{sf}_X(x)\), the survival function (sf) of a random variable \(X\), following the distribution of Pearson’s rho (under \(H_0\)):

\[\text{sf}_X(r) = F_{\text{StudentT}}\left(-t, N-2\right), \quad \text{where } t = r \sqrt{\frac{N-2}{1-r^2}},\]

and \(F_{\text{StudentT}}\left(t, N-2,\right)\) is the cdf of the t-distribution with \(N-2\) degrees of freedom.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print (" sf: ", pearson_rho(mu, sigma).pdf(x))
sf: 6.3563523462564525615615615614561356E-20

dist_pearson_rho.qtf(q)

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following the distribution of Pearson’s rho (under \(H_0\)):

\[\text{qtf}_X(q) = \frac{t_{\alpha, N-2}}{\sqrt{N-2+t_{\alpha, N-2}^2}},\]

where \(t_{\alpha, N-2}\) is the quantile function of the central t-distribution with \(N-2\) degrees of freedom.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("qtf: ", pearson_rho(mu, sigma).qtf(q))
qtf: 6.3563523462564525615615615614561356E+00

dist_pearson_rho.isf(q)

Returns \(\text{isf}_X(q)\), the inverse survival function (isf) of a random variable \(X\), following the distribution of Pearson’s rho (under \(H_0\)):

\[\text{isf}_X(q) = \frac{-t_{\alpha, N-2}}{\sqrt{N-2+t_{\alpha, N-2}^2}},\]

where \(t_{\alpha, N-2}\) is the quantile function of the central t-distribution with \(N-2\) degrees of freedom.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("isf: ", pearson_rho(mu, sigma).isf(q))
6.3563523462564525615615615614561356E+00

dist_pearson_rho.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following the distribution of Pearson’s rho (under \(H_0\)):

\[C_X(t) = \Gamma(\tfrac{1}{2}(N-1)) \cdot 2^{(N-3)/2} \cdot t^{-(N-3)/2} \cdot J_{(N-3)/2}(t),\]

where \(J_{\nu}(t)\) is the Bessel function of the first kind of order \(\nu\).

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3;
>>> print ("c_x: ", pearson_rho(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_pearson_rho.m_x(t)

Returns \(M_X(t)\), the moment generating function of a random variable \(X\), following the distribution of Pearson’s rho (under \(H_0\)):

\[M_X(t) = \Gamma(\tfrac{1}{2}(N-1)) \cdot 2^{(N-3)/2} \cdot t^{-(N-3)/2} \cdot I_{(N-3)/2}(t),\]

where \(I_{\nu}(t)\) is the modified Bessel function of the second kind of order \(\nu\).

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3;
>>> print ("c_x: ", pearson_rho(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_pearson_rho.k_x(t, k=0)

Returns \(K_X(t)\), the cumulant generating function of a random variable \(X\), following the distribution of Pearson’s rho (under \(H_0\)):

\[K_X(t) = \log \left( \Gamma(\tfrac{1}{2}(N-1)) \cdot 2^{(N-3)/2} \cdot t^{-(N-3)/2} \cdot I_{(N-3)/2}(t) \right),\]

where \(I_{\nu}(t)\) is the modified Bessel function of the second kind of order \(\nu\).

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3; k = 6;
>>> print ("c_x: ", pearson_rho(mu, sigma).k_x(t, k))
6.3563523462564525615615615614561356E+00

dist_pearson_rho.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following the distribution of Pearson’s rho (under \(H_0\)):

All odd moments are zero. For the even moments we have

\[\mu_X(r) = \tfrac{1}{2} B\left( \tfrac{1}{2} (r+1), \tfrac{1}{2} (N-2) \right)\]

where \(B(a,b)\) is the beta function.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", pearson_rho(mu, sigma).moments(k))
6.3563523462564525615615615614561356E+00

dist_pearson_rho.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following the distribution of Pearson’s rho (under \(H_0\)). The cumulants are calculated from the moments.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", pearson_rho(mu, sigma).cumulants(k))
6.3563523462564525615615615614561356E+00