Doubly non-central Student \(t\) distribution#

class ctx.dist_student_t_2nc(n, delta, theta)#

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

The doubly non-central Student \(t\) distribution is a continuous probability distribution with \(n>0\) degrees of freedom, noncentrality parameters \(\delta\) and \(\theta\), and support interval \((-\infty, +\infty)\). See also Broda and Paolella [167], Kocherlakota and Kocherlakota [416], Paolella [478], Paolella [479], Gessner [363],

dist_student_t_2nc.pdf(x)#

Returns \(\text{pdf}_X(x)\), the probability density function (pdf) of a random variable \(X\), following a doubly non-central Student t distribution:

\[\text{pdf}_X(x) = f_{t''}(t;n;\mu,\theta) = \sum_{i=0}^{\infty} \omega_{i,\theta} s_{i,n} f_{t'}(s_{i,n} x;n+2i,\mu),\]

where \(f_{t'}(\cdot)\) denotes the PDF of the singly noncentral \(t\)-distribution, and

\[\omega_{i,\theta} = \frac{\exp(-\theta/2)(\theta/2)^i}{i!} \quad \text{and} \quad s_{i,n}=\sqrt{\frac{n+2i}{n}}.\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("pdf: ", student_t_2nc(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_student_t_2nc.cdf(x)#

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following a doubly non-central Student t distribution:

\[\text{cdf}_X(x) = F_{t''}(x;n;\mu,\theta) = \int_{0}^{x} f_{t''}(x;n;\mu,\theta) \mathrm{d} t = \sum_{i=0}^{\infty} \omega_{i,\theta} s_{i,n} F_{t'}(s_{i,n} x;n+2i,\mu),\]

where \(F_{t'}(\cdot)\) denotes the CDF of the singly noncentral \(t\)-distribution, and

\[\omega_{i,\theta} = \frac{\exp(-\theta/2)(\theta/2)^i}{i!} \quad \text{and} \quad s_{i,n}=\sqrt{\frac{n+2i}{n}}.\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("cdf: ", student_t_2nc(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_student_t_2nc.sf(x)#

Returns \(\text{sf}_X(x)\), the survival function (sf) of a random variable \(X\), following a doubly non-central Student t distribution:

\[\text{sf}_X(x) = 1 - \text{cdf}_X(x) = \int_{x}^{\infty} \text{pdf}_X(x) \mathrm{d} t.\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print (" sf: ", student_t_2nc(mu, sigma).pdf(x))
sf: 6.3563523462564525615615615614561356E-20

dist_student_t_2nc.qtf(q)#

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following a doubly non-central Student t distribution:

There is no known explicit form for the quantile function \(\text{cdf}^{-1}_X(x)\): It is computed using Newton iterations with starting values from a singly noncentral \(t\) approximation.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("qtf: ", student_t_2nc(mu, sigma).qtf(q))
qtf: 6.3563523462564525615615615614561356E+00

dist_student_t_2nc.isf(q)#

Returns \(\text{isf}_X(q)\), the inverse survival function (isf) of a random variable \(X\), following a doubly non-central Student t distribution:

\[\text{isf}_X(q) = \text{qtf}_X(1-q) = \text{cdf}^{-1}_X(1-q).\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("isf: ", student_t_2nc(mu, sigma).isf(q))
6.3563523462564525615615615614561356E+00

dist_student_t_2nc.c_x(t)#

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following a doubly non-central Student t distribution:

\[C_X(t) = \int_{-\infty}^{\infty} e^{i tx} \text{pdf}_X(x) \mathrm{d} x\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3;
>>> print ("c_x: ", student_t_2nc(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_student_t_2nc.m_x(t)#: Returns NaN, since the moment generating function does not exist.

dist_student_t_2nc.k_x(t, k=0)#: Returns NaN, since the cumulant generating function does not exist.

dist_student_t_2nc.moments(k)#

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a doubly non-central Student t distribution. The rth moment only exists for \(n > r\) and is given by

\[\mu'_X(r) = \left({\tfrac{1}{2}n}\right)^{r/2} \frac{\Gamma\left(\tfrac{1}{2}(n-r)\right)}{\Gamma\left(\tfrac{1}{2}n\right)} \times {}_1F_1(\tfrac{1}{2}r, \tfrac{1}{2}n, -\tfrac{1}{2}\theta) \times \sum_{i=0}^{\lfloor r/2 \rfloor} { \binom{r}{2i} \frac{(2i)!} {2^i i!}} \delta^{r-2i},\]

where \({}_1F_1(\cdot)\) denotes Kummer’s confluent hypergeometric function.

Doubly non-central Student t distribution

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Doubly non-central Student \(t\) distribution#