naginterfaces.library.stat.prob_​multi_​students_​t

naginterfaces.library.stat.prob_multi_students_t(tail, a, b, nu, delta, iscov, rc, epsabs=0.0, epsrel=0.001, numsub=350, nsampl=8, fmax=None)[source]

prob_multi_students_t returns a probability associated with a multivariate Student’s -distribution.

For full information please refer to the NAG Library document for g01hd

https://www.nag.com/numeric/nl/nagdoc_28.6/flhtml/g01/g01hdf.html

Parameters
tailstr, length 1, array-like, shape

Defines the calculated probability, set to:

If the th lower limit is negative infinity.

If the th upper limit is infinity.

If both and are finite.

afloat, array-like, shape

, for , the lower integral limits of the calculation.

If , is not referenced and the th lower limit of integration is .

bfloat, array-like, shape

, for , the upper integral limits of the calculation.

If , is not referenced and the th upper limit of integration is .

nufloat

, the degrees of freedom.

deltafloat, array-like, shape

the noncentrality parameter for the th dimension, for ; set for the central probability.

iscovint

Set if the covariance matrix is supplied and if the correlation matrix is supplied.

rcfloat, array-like, shape

The lower triangle of either the covariance matrix (if ) or the correlation matrix (if ). In either case the array elements corresponding to the upper triangle of the matrix need not be set.

epsabsfloat, optional

, the absolute accuracy requested in the approximation. If is negative, the absolute value is used.

epsrelfloat, optional

, the relative accuracy requested in the approximation. If is negative, the absolute value is used.

numsubint, optional

If quadrature is used, the number of sub-intervals used by the quadrature algorithm; otherwise is not referenced.

nsamplint, optional

If quadrature is used, is not referenced; otherwise is the number of samples used to estimate the error in the approximation.

fmaxNone or int, optional

Note: if this argument is None then a default value will be used, determined as follows: .

If a number theoretic approach is used, the maximum number of evaluations for each integrand function.

Returns
pfloat

The probability associated with the multivariate Student’s -distribution.

rcfloat, ndarray, shape

The strict upper triangle of contains the correlation matrix used in the calculations.

errestfloat

An estimate of the error in the calculated probability.

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: , or .

(errno )

On entry, .

Constraint: for a central probability.

(errno )

On entry, .

Constraint: degrees of freedom .

(errno )

On entry, .

Constraint: or .

(errno )

On entry, the information supplied in is invalid.

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

Notes

A random vector that follows a Student’s -distribution with degrees of freedom and covariance matrix has density:

and probability given by:

The method of calculation depends on the dimension and degrees of freedom . The method of Dunnett and Sobel (1954) is used in the bivariate case if is a whole number. A Plackett transform followed by quadrature method is adopted in other bivariate cases and trivariate cases. In dimensions higher than three a number theoretic approach to evaluating multidimensional integrals is adopted.

Error estimates are supplied as the published accuracy in the Dunnett and Sobel (1954) case, a Monte Carlo standard error for multidimensional integrals, and otherwise the quadrature error estimate.

A parameter allows for non-central probabilities. The number theoretic method is used if any is nonzero.

In cases other than the central bivariate with whole , prob_multi_students_t attempts to evaluate probabilities within a requested accuracy , for an approximate integral value , absolute accuracy and relative accuracy .

References

Dunnett, C W and Sobel, M, 1954, A bivariate generalization of Student’s -distribution, with tables for certain special cases, Biometrika (41), 153–169

Genz, A and Bretz, F, 2002, Methods for the computation of multivariate -probabilities, Journal of Computational and Graphical Statistics ((11)), 950–971