NAG Library Function Document

1Purpose

nag_multi_students_t (g01hdc) returns a probability associated with a multivariate Student's $t$-distribution.

2Specification

 #include #include
 double nag_multi_students_t (Integer n, const Nag_TailProbability tail[], const double a[], const double b[], double nu, const double delta[], Nag_Boolean iscov, double rc[], Integer pdrc, double epsabs, double epsrel, Integer numsub, Integer nsampl, Integer fmax, double *errest, NagError *fail)

3Description

A random vector $x\in {ℝ}^{n}$ that follows a Student's $t$-distribution with $\nu$ degrees of freedom and covariance matrix $\Sigma$ has density:
 $Γ ν+n / 2 Γ ν/2 νn/2 πn/2 Σ 1/2 1+ 1ν xT Σ-1x ν+n / 2 ,$
and probability $p$ given by:
 $p = Γ ν+n / 2 Γ ν/2 Σ πνn ∫ a1 b1 ∫ a2 b2 ⋯ ∫ an bn 1+ xT Σ-1x/ν - ν+n/2 dx .$
The method of calculation depends on the dimension $n$ and degrees of freedom $\nu$. The method of Dunnet and Sobel is used in the bivariate case if $\nu$ 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 $\delta$ allows for non-central probabilities. The number theoretic method is used if any $\delta$ is nonzero.
In cases other than the central bivariate with whole $\nu$, nag_multi_students_t (g01hdc) attempts to evaluate probabilities within a requested accuracy $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\epsilon }_{a},{\epsilon }_{r}×I\right)$, for an approximate integral value $I$, absolute accuracy ${\epsilon }_{a}$ and relative accuracy ${\epsilon }_{r}$.
Dunnet C W and Sobel M (1954) A bivariate generalization of Student's $t$-distribution, with tables for certain special cases Biometrika 41 153–169
Genz A and Bretz F (2002) Methods for the computation of multivariate $t$-probabilities Journal of Computational and Graphical Statistics (11) 950–971

5Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of dimensions.
Constraint: $1<{\mathbf{n}}\le 1000$.
2:    $\mathbf{tail}\left[{\mathbf{n}}\right]$const Nag_TailProbabilityInput
On entry: defines the calculated probability, set ${\mathbf{tail}}\left[i-1\right]$ to:
${\mathbf{tail}}\left[i-1\right]=\mathrm{Nag_LowerTail}$
If the $i$th lower limit ${a}_{i}$ is negative infinity.
${\mathbf{tail}}\left[i-1\right]=\mathrm{Nag_UpperTail}$
If the $i$th upper limit ${b}_{i}$ is infinity.
${\mathbf{tail}}\left[i-1\right]=\mathrm{Nag_Central}$
If both ${a}_{i}$ and ${b}_{i}$ are finite.
Constraint: ${\mathbf{tail}}\left[\mathit{i}-1\right]=\mathrm{Nag_LowerTail}$, $\mathrm{Nag_UpperTail}$ or $\mathrm{Nag_Central}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3:    $\mathbf{a}\left[{\mathbf{n}}\right]$const doubleInput
On entry: ${a}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, the lower integral limits of the calculation.
If ${\mathbf{tail}}\left[i-1\right]=\mathrm{Nag_LowerTail}$, ${\mathbf{a}}\left[i-1\right]$ is not referenced and the $i$th lower limit of integration is $-\infty$.
4:    $\mathbf{b}\left[{\mathbf{n}}\right]$const doubleInput
On entry: ${b}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, the upper integral limits of the calculation.
If ${\mathbf{tail}}\left[i-1\right]=\mathrm{Nag_UpperTail}$, ${\mathbf{b}}\left[i-1\right]$ is not referenced and the $i$th upper limit of integration is $\infty$.
Constraint: if ${\mathbf{tail}}\left[i-1\right]=\mathrm{Nag_Central}$, ${\mathbf{b}}\left[i-1\right]>{\mathbf{a}}\left[i-1\right]$.
5:    $\mathbf{nu}$doubleInput
On entry: $\nu$, the degrees of freedom.
Constraint: ${\mathbf{nu}}>0.0$.
6:    $\mathbf{delta}\left[{\mathbf{n}}\right]$const doubleInput
On entry: ${\mathbf{delta}}\left[\mathit{i}-1\right]$ the noncentrality parameter for the $\mathit{i}$th dimension, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$; set ${\mathbf{delta}}\left[i-1\right]=0$ for the central probability.
7:    $\mathbf{iscov}$Nag_BooleanInput
On entry: set ${\mathbf{iscov}}=\mathrm{Nag_TRUE}$ if the covariance matrix is supplied and ${\mathbf{iscov}}=\mathrm{Nag_FALSE}$ if the correlation matrix is supplied.
8:    $\mathbf{rc}\left[{\mathbf{n}}×{\mathbf{pdrc}}\right]$doubleInput/Output
Note: the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{rc}}\left[\left(i-1\right)×{\mathbf{pdrc}}+j-1\right]$.
On entry: the lower triangle of either the covariance matrix (if ${\mathbf{iscov}}=\mathrm{Nag_TRUE}$) or the correlation matrix (if ${\mathbf{iscov}}=\mathrm{Nag_FALSE}$). 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:    $\mathbf{pdrc}$IntegerInput
On entry: the stride separating matrix column elements in the array rc.
Constraint: ${\mathbf{pdrc}}\ge {\mathbf{n}}$.
10:  $\mathbf{epsabs}$doubleInput
On entry: ${\epsilon }_{a}$, the absolute accuracy requested in the approximation. If epsabs is negative, the absolute value is used.
Suggested value: $0.0$.
11:  $\mathbf{epsrel}$doubleInput
On entry: ${\epsilon }_{r}$, the relative accuracy requested in the approximation. If epsrel is negative, the absolute value is used.
Suggested value: $0.001$.
12:  $\mathbf{numsub}$IntegerInput
On entry: if quadrature is used, the number of sub-intervals used by the quadrature algorithm; otherwise numsub is not referenced.
Suggested value: $350$.
Constraint: if referenced, ${\mathbf{numsub}}>0$.
13:  $\mathbf{nsampl}$IntegerInput
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: $8$.
Constraint: if referenced,${\mathbf{nsampl}}>0$.
14:  $\mathbf{fmax}$IntegerInput
On entry: if a number theoretic approach is used, the maximum number of evaluations for each integrand function.
Suggested value: $1000×{\mathbf{n}}$.
Constraint: if referenced,${\mathbf{fmax}}\ge 1$.
15:  $\mathbf{errest}$double *Output
On exit: an estimate of the error in the calculated probability.
16:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_ARRAY_SIZE
On entry, ${\mathbf{pdrc}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdrc}}\ge {\mathbf{n}}$.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{fmax}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{fmax}}\ge 1$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: $1<{\mathbf{n}}\le 1000$.
On entry, ${\mathbf{nsampl}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nsampl}}\ge 1$.
On entry, ${\mathbf{numsub}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{numsub}}\ge 1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_INVALID_ARRAY
On entry, the information supplied in rc is invalid.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL
On entry, ${\mathbf{nu}}=〈\mathit{\text{value}}〉$.
Constraint: degrees of freedom ${\mathbf{nu}}>0.0$.
NE_REAL_2
On entry, $k=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{b}}\left[k-1\right]>{\mathbf{a}}\left[k-1\right]$ for a central probability.

7Accuracy

An estimate of the error in the calculation is given by the value of errest on exit.

8Parallelism and Performance

nag_multi_students_t (g01hdc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_multi_students_t (g01hdc) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

None.

10Example

This example prints two probabilities from the Student's $t$-distribution.

10.1Program Text

Program Text (g01hdce.c)

10.2Program Data

Program Data (g01hdce.d)

10.3Program Results

Program Results (g01hdce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017