NAG FL Interface
g01hdf (prob_multi_students_t)
1
Purpose
g01hdf returns a probability associated with a multivariate Student's -distribution.
2
Specification
Fortran Interface
Function g01hdf ( |
n, tail, a, b, nu, delta, iscov, rc, ldrc, epsabs, epsrel, numsub, nsampl, fmax, errest, ifail) |
Real (Kind=nag_wp) |
:: |
g01hdf |
Integer, Intent (In) |
:: |
n, iscov, ldrc, numsub, nsampl, fmax |
Integer, Intent (Inout) |
:: |
ifail |
Real (Kind=nag_wp), Intent (In) |
:: |
a(n), b(n), nu, delta(n), epsabs, epsrel |
Real (Kind=nag_wp), Intent (Inout) |
:: |
rc(ldrc,n) |
Real (Kind=nag_wp), Intent (Out) |
:: |
errest |
Character (1), Intent (In) |
:: |
tail(n) |
|
C Header Interface
#include <nag.h>
double |
g01hdf_ (const Integer *n, const char tail[], const double a[], const double b[], const double *nu, const double delta[], const Integer *iscov, double rc[], const Integer *ldrc, const double *epsabs, const double *epsrel, const Integer *numsub, const Integer *nsampl, const Integer *fmax, double *errest, Integer *ifail, const Charlen length_tail) |
|
C++ Header Interface
#include <nag.h> extern "C" {
double |
g01hdf_ (const Integer &n, const char tail[], const double a[], const double b[], const double &nu, const double delta[], const Integer &iscov, double rc[], const Integer &ldrc, const double &epsabs, const double &epsrel, const Integer &numsub, const Integer &nsampl, const Integer &fmax, double &errest, Integer &ifail, const Charlen length_tail) |
}
|
The routine may be called by the names g01hdf or nagf_stat_prob_multi_students_t.
3
Description
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 Dunnet and Sobel 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 Dunnet and Sobel 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 , g01hdf attempts to evaluate probabilities within a requested accuracy , for an approximate integral value , absolute accuracy and relative accuracy .
4
References
Dunnet 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
5
Arguments
-
1:
– Integer
Input
-
On entry: , the number of dimensions.
Constraint:
.
-
2:
– Character(1) array
Input
-
On entry: 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.
Constraint:
, or , for .
-
3:
– Real (Kind=nag_wp) array
Input
-
On entry:
, for
, the lower integral limits of the calculation.
If , is not referenced and the th lower limit of integration is .
-
4:
– Real (Kind=nag_wp) array
Input
-
On entry:
, for
, the upper integral limits of the calculation.
If , is not referenced and the th upper limit of integration is .
Constraint:
if , .
-
5:
– Real (Kind=nag_wp)
Input
-
On entry: , the degrees of freedom.
Constraint:
.
-
6:
– Real (Kind=nag_wp) array
Input
-
On entry: the noncentrality parameter for the th dimension, for ; set for the central probability.
-
7:
– Integer
Input
-
On entry: set if the covariance matrix is supplied and if the correlation matrix is supplied.
Constraint:
or .
-
8:
– Real (Kind=nag_wp) array
Input/Output
-
On entry: 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.
On exit: the strict upper triangle of
rc contains the correlation matrix used in the calculations.
-
9:
– Integer
Input
-
On entry: the first dimension of the array
rc as declared in the (sub)program from which
g01hdf is called.
Constraint:
.
-
10:
– Real (Kind=nag_wp)
Input
-
On entry:
, the absolute accuracy requested in the approximation. If
epsabs is negative, the absolute value is used.
Suggested value:
.
-
11:
– Real (Kind=nag_wp)
Input
-
On entry:
, the relative accuracy requested in the approximation. If
epsrel is negative, the absolute value is used.
Suggested value:
.
-
12:
– Integer
Input
-
On entry: if quadrature is used, the number of sub-intervals used by the quadrature algorithm; otherwise
numsub is not referenced.
Suggested value:
.
Constraint:
if referenced, .
-
13:
– Integer
Input
-
On entry: if quadrature is used,
nsampl is not referenced; otherwise
nsampl is the number of samples used to estimate the error in the approximation.
Suggested value:
.
Constraint:
if referenced,.
-
14:
– Integer
Input
-
On entry: if a number theoretic approach is used, the maximum number of evaluations for each integrand function.
Suggested value:
.
Constraint:
if referenced,.
-
15:
– Real (Kind=nag_wp)
Output
-
On exit: an estimate of the error in the calculated probability.
-
16:
– Integer
Input/Output
-
On entry:
ifail must be set to
,
. If you are unfamiliar with this argument you should refer to
Section 4 in the Introduction to the NAG Library FL Interface for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
.
When the value is used it is essential to test the value of ifail on exit.
On exit:
unless the routine detects an error or a warning has been flagged (see
Section 6).
6
Error Indicators and Warnings
If on entry
or
, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
-
On entry, .
Constraint: .
-
On entry, .
Constraint: , or .
-
On entry, .
Constraint: for a central probability.
-
On entry, .
Constraint: degrees of freedom .
-
On entry, .
Constraint: or .
-
On entry, the information supplied in
rc is invalid.
-
On entry, and .
Constraint: .
-
On entry, .
Constraint: .
-
On entry, .
Constraint: .
-
On entry, .
Constraint: .
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See
Section 9 in the Introduction to the NAG Library FL Interface for further information.
7
Accuracy
An estimate of the error in the calculation is given by the value of
errest on exit.
8
Parallelism and Performance
g01hdf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g01hdf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
None.
10
Example
This example prints two probabilities from the Student's -distribution.
10.1
Program Text
10.2
Program Data
10.3
Program Results