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

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

prob_students_t_noncentral returns the lower tail probability for the noncentral Student’s -distribution.

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

https://www.nag.com/numeric/nl/nagdoc_28.4/flhtml/g01/g01gbf.html

Parameters
tfloat

, the deviate from the Student’s -distribution with degrees of freedom.

dffloat

, the degrees of freedom of the Student’s -distribution.

deltafloat

, the noncentrality parameter of the Students -distribution.

tolfloat, optional

The absolute accuracy required by you in the results. If prob_students_t_noncentral is entered with greater than or equal to or less than (see machine.precision), the value of 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 -distribution.

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

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

Warns
NagAlgorithmicWarning
(errno )

One of the series has failed to converge with and . Reconsider the requested tolerance and/or the maximum number of iterations.

(errno )

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

Notes

The lower tail probability of the noncentral Student’s -distribution with degrees of freedom and noncentrality parameter , , is defined by

with

The probability is computed in one of two ways.

  1. When , the relationship to the normal is used:

  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 -distributions, Biometrika (51), 451–458