Kendall’s tau distribution, continuous data

class ctx.dist_kendall_tau(N)

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

The distribution of \(K_N = \tfrac{1}{4} N(N-1)(1-\tau)\) is a discrete (lattice) probability distribution with sample size \(N \ge 2\) and the support interval \((0, N(N-1)/2)\). See also Wikipedia [1274], Noether [447], vandeWiel [860], Robillard [507].

Let \((X_1, Y_1),...,(X_N, Y_N)\) be a sample of \(N\) pairs of observations. The Kendall rank correlation coefficient \(\tau\) is defined as

\[\tau = 1 - \frac{2 K_N}{N(N-1)/2}\]

where \(K_N\) is the number of inversions: the number of pairs \(\{(X_i, Y_i),(X_j, Y_j)\}\) such that \(X_i < X_j\) and \(Y_i > Y_j\) for \(i < j\), where \(i = 1, \ldots n-1\) and \(j = 2,\ldots,n\). \(K_N\) can assume values between \(0\) and \(N(N-1)/2\), whereas \(\tau\) can assume values between \(-1\) and \(+1\).

dist_kendall_tau.pmf(x)

Returns \(\text{pmf}_X(x)\), the probability mass function (pmf) of a random variable \(X\), following the distribution of Kendall’s \(K_N\).

\[\text{pmf}_X(x) = \sum_{j=x}^{N(N-1)/2} (-1)^{x+j} \binom{j}{x} \frac{\mu'_{[j]}}{j!},\]

where \(\mu'_{[j]}\) is the \(j^{\text{th}}\) factorial moment.

The factorial moments are calculated from the cumulants (see factorial_moments_from_cumulants()).

The pmf can also be calculated from the characteristic function \(C_X(t)\):

\[\text{pmf}(x) = \frac{1}{\pi} \int_{0}^{\pi} \Re \left( e^{-itx} C_X(t) \right) \mathrm{d} t,\]

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

dist_kendall_tau.cdf(x)

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

\[\text{cdf}_X(x) = \sum_{j=x}^{N(N-1)/2} (-1)^{x+j} \binom{j-1}{x-1} \frac{\mu'_{[j]}}{j!},\]

where \(\mu'_{[j]}\) is the \(j^{\text{th}}\) factorial moment.

The cdf can also be calculated from the characteristic function \(C_X(t)\):

\[\text{cdf}(x) = \frac{1}{\pi} \int_{0}^{\pi} \Re \left( C_X(t) \sum_{z=0}^x e^{-itz} \right) \mathrm{d} t.\]

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

dist_kendall_tau.sf(x)

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

\[\text{sf}_X(x) = 1-\sum_{j=x}^{N(N-1)/2} (-1)^{x+j} \binom{j-1}{x-1} \frac{\mu'_{[j]}}{j!},\]

where \(\mu'_{[j]}\) is the \(j^{\text{th}}\) factorial moment.

The cdf can also be calculated from the characteristic function \(C_X(t)\):

\[\text{sf}(x) = 1-\text{cdf}(x) = \frac{1}{\pi} \int_{0}^{\pi} \Re \left( C_X(t) \sum_{z=x+1}^{N(N-1)/2} e^{-itz} \right) \mathrm{d} t.\]

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

dist_kendall_tau.qtf(q)

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following the distribution of Kendall’s \(K_N\). There is no closed form for the qtf: It is computed using the Brent algorithm with starting values from a Cornish-Fisher or Jensen approximation.

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

dist_kendall_tau.isf(q)

Returns \(\text{isf}_X(q)\), the inverse survival function (isf) of a random variable \(X\), following the distribution of Kendall’s \(K_N\). There is no closed form for the isf: It is computed using the Brent algorithm with starting values from a Cornish-Fisher or Jensen approximation

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

dist_kendall_tau.g_x(t)

Returns \(G_X(t)\), the probability generating function of a random variable \(X\), following the distribution of Kendall’s \(K_N\):

\[G_X(t) = \frac{1}{N!} \prod_{k=1}^{N} \frac{(t^k-1)}{(t-1)}.\]

See also: v.d.Wiel, p. 16, equ. 2.1

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

dist_kendall_tau.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following the distribution of Kendall’s \(K_N\):

\[C_X(t) = \frac{1}{N!} \prod_{k=1}^{N} \frac{\exp(it \cdot k)-1}{\exp(it)-1}.\]

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

dist_kendall_tau.m_x(t)

Returns \(M_X(t)\), the moment generating function of a random variable \(X\), following the distribution of Kendall’s \(K_N\):

\[M_X(t) = \frac{1}{N!} \prod_{k=1}^{N} \frac{\exp(t \cdot k)-1}{\exp(t)-1}.\]

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

dist_kendall_tau.k_x(t, k=0)

Returns \(K_X(t)\), the cumulant generating function of a random variable \(X\), following the distribution of Kendall’s \(K_N\):

\[K_X(t) = -\log(N!) + \sum_{k=1}^{N} \log \left( \frac{\exp(t \cdot k)-1}{\exp(t)-1} \right).\]

\(K_X(t)\), 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 Kendall tau distribution, are defined as

\[K_X(t) = \sum_{j=2}^{k} \log \left( \frac{1}{N_j + 1} \frac{1-\exp((N_j+1)t)}{1-\exp(t)} \right).\]

\[K_X^{(1)}(s) = \sum_{r=1}^{m} \left( \frac{n+r}{1-\exp((n+r)s)} - \frac{n(\exp(r \cdot s)+n+r)}{1-\exp(r \cdot s)} \right),\]

\[K_X^{(j)}(s) = (-1)^j \sum_{r=1}^{m} \sum_{i=0}^{1} (-1)^i \cdot t_i^j \cdot \sum_{k=1}^{j} c(j-2,k) \cdot z_i^j, \quad j \ge 2, \quad \text{where}\]

\[\begin{split}z_i = \frac{1}{1-\exp(t_i \cdot s)}, \quad t_i = \begin{cases} n+r, & i=0,\\ r & i=1, \end{cases}\end{split}\]

and the coefficients \(c(i,j)\) are calculated recursively, with \(c(0,1) = c(0,2)=c(i,1)=1\), and

\[c(i,j) = (j-1) \cdot c(i-1,j-1) + j \cdot c(i-1,j), \quad j \ge 2.\]

The saddlepoint \(s\) is determined numerically using Newton iterations, with a starting value of \(s=0.1\).

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

dist_kendall_tau.moments(k)

Returns the first \(j\) moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following the distribution of Kendall’s \(K_N\) (Wikipedia). The moments are calculated from the cumulants.

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

dist_kendall_tau.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following the distribution of Kendall’s \(K_N\). The cumulants of \(T_N\) are given by

\[\kappa_{2j}(T_N) = \frac{B_{2j}}{2j} \sum_{s=1}^N s^{2j} = \frac{B_{2j}}{2j} \left[ \frac{B_{2j+1}(N+1)-B_{2j+1}}{2j+1} - N \right], \quad \text{and}\]

\[\kappa_{2j+1}(W_N) = 0, \quad \text{for } j \geq 1.\]

In particular, \(\kappa_1(T_N)=N(N-1)/4\), and \(\kappa_2(T_N)= N(N-1)(2N+5)/72\), and \(B_{2j}\) and \(B_{2j}(x)\) are the Bernoulli numbers and polynomials, respectively, of degree \(2j\).

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

Approximations

ctx.kendall_ft(x, n, results='cdf')

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

Calculates the pdf, cdf and sf from the characteristic function (see pmf_from_cf_lattice() and cdf_from_cf_lattice()).

ctx.kendall_tau_ecf(x, f, rho, omega)

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

Calculates the Edgeworth approximation to the pdf.

ctx.kendall_tau_ecf_inv(q, f, rho, omega)

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

Calculates the Cornish-Fisher approximation to the qtf and isf.

ctx.kendall_tau_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.

The saddlepoint \(s\) is determined numerically using Newton iterations, with a starting value of \(s=0.1\).

ctx.kendall_tau_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.