naginterfaces.library.stat.prob_​students_​t_​noncentral

naginterfaces.library.stat.prob_students_t_noncentral(t, df, delta, tol=0.0, maxit=100)

prob_students_t_noncentral returns the lower tail probability for the noncentral Student’s $t$-distribution.

For full information please refer to the NAG Library document for g01gb

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g01/g01gbf.html

Parameters

tfloat

$t$, the deviate from the Student’s $t$-distribution with $\nu$ degrees of freedom.

dffloat

$\nu$, the degrees of freedom of the Student’s $t$-distribution.

deltafloat

$\delta$, the noncentrality parameter of the Students $t$-distribution.

tolfloat, optional

The absolute accuracy required by you in the results. If prob_students_t_noncentral is entered with $tol$ greater than or equal to $1.0$ or less than $10\times \text{machine precision}$ (see machine.precision), the value of $10\times \text{machine precision}$ is used instead.

maxitint, optional

The maximum number of terms that are used in each of the summations.

Returns

pfloat

The lower tail probability for the noncentral Student’s $t$-distribution.

Raises

NagValueError

(errno $1$)

On entry, $df=⟨value⟩$.

Constraint: $df\ge 1.0$.

(errno $2$)

On entry, $maxit=⟨value⟩$.

Constraint: $maxit\ge 1$.

(errno $4$)

Unable to calculate the probability as it is too close to zero or one.

Warns

NagAlgorithmicWarning

(errno $3$)

One of the series has failed to converge with $maxit=⟨value⟩$ and $tol=⟨value⟩$. Reconsider the requested tolerance and/or the maximum number of iterations.

(errno $4$)

The probability is too close to $0$ or $1$. The returned value should be a reasonable estimate of the true value.

Notes

The lower tail probability of the noncentral Student’s $t$-distribution with $\nu$ degrees of freedom and noncentrality parameter $\delta$, $P(T\le t:\nu \text{;}\delta )$, is defined by

$$P(T\le t:\nu \text{;}\delta )=C_{\nu }{\int }_0^{\infty }\left(\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\alpha u-\delta }{e}^{-x^2/2}dx\right)u^{\nu -1}{e}^{-u^2/2}du\text{,}\nu >0.0$$

with

$$C_{\nu }=\frac{1}{\Gamma \left(\frac{1}{2}\nu \right)2^{(\nu -2)/2}}\text{,}\alpha =\frac{t}{\sqrt{\nu }}\text{.}$$

The probability is computed in one of two ways.

1. When $t=0.0$, the relationship to the normal is used:

   $$P(T\le t:\nu \text{;}\delta )=\frac{1}{\sqrt{2\pi }}{\int }_{\delta }^{\infty }{e}^{-u^2/2}du\text{.}$$

2. Otherwise the series expansion described in Equation 9 of Amos (1964) is used. This involves the sums of confluent hypergeometric functions, the terms of which are computed using recurrence relationships.

References

Amos, D E, 1964, Representations of the central and non-central $t$ -distributions, Biometrika (51), 451–458