NAG FL Interfaceg01sbf (prob_​students_​t_​vector)

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1Purpose

g01sbf returns a number of one or two tail probabilities for the Student's $t$-distribution with real degrees of freedom.

2Specification

Fortran Interface
 Subroutine g01sbf ( tail, lt, t, ldf, df, p,
 Integer, Intent (In) :: ltail, lt, ldf Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ivalid(*) Real (Kind=nag_wp), Intent (In) :: t(lt), df(ldf) Real (Kind=nag_wp), Intent (Out) :: p(*) Character (1), Intent (In) :: tail(ltail)
#include <nag.h>
 void g01sbf_ (const Integer *ltail, const char tail[], const Integer *lt, const double t[], const Integer *ldf, const double df[], double p[], Integer ivalid[], Integer *ifail, const Charlen length_tail)
The routine may be called by the names g01sbf or nagf_stat_prob_students_t_vector.

3Description

The lower tail probability for the Student's $t$-distribution with ${\nu }_{i}$ degrees of freedom, $P\left({T}_{i}\le {t}_{i}:{\nu }_{i}\right)$ is defined by:
 $P( Ti ≤ ti :νi) = Γ ((νi+1)/2) πνi Γ(νi/2) ∫ -∞ ti [1+Ti2νi] -(νi+1) / 2 dTi , νi≥1 .$
Computationally, there are two situations:
1. (i)when ${\nu }_{i}<20$, a transformation of the beta distribution, ${P}_{{\beta }_{i}}\left({B}_{i}\le {\beta }_{i}:{a}_{i},{b}_{i}\right)$ is used
 $P( Ti ≤ ti :νi) = 12 Pβi( Bi≤ νi νi+ti2 :νi/2,12) when ​ ti<0.0$
or
 $P( Ti ≤ ti :νi) = 12 + 12 Pβi( Bi ≥ νi νi + ti2 :νi/2,12) when ​ ti>0.0 ;$
2. (ii)when ${\nu }_{i}\ge 20$, an asymptotic normalizing expansion of the Cornish–Fisher type is used to evaluate the probability, see Hill (1970).
The input arrays to this routine are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section 2.6 in the G01 Chapter Introduction for further information.

4References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth
Hill G W (1970) Student's $t$-distribution Comm. ACM 13(10) 617–619

5Arguments

1: $\mathbf{ltail}$Integer Input
On entry: the length of the array tail.
Constraint: ${\mathbf{ltail}}>0$.
2: $\mathbf{tail}\left({\mathbf{ltail}}\right)$Character(1) array Input
On entry: indicates which tail the returned probabilities should represent. For , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lt}},{\mathbf{ldf}}\right)$:
${\mathbf{tail}}\left(j\right)=\text{'L'}$
The lower tail probability is returned, i.e., ${p}_{i}=P\left({T}_{i}\le {t}_{i}:{\nu }_{i}\right)$.
${\mathbf{tail}}\left(j\right)=\text{'U'}$
The upper tail probability is returned, i.e., ${p}_{i}=P\left({T}_{i}\ge {t}_{i}:{\nu }_{i}\right)$.
${\mathbf{tail}}\left(j\right)=\text{'C'}$
The two tail (confidence interval) probability is returned, i.e., ${p}_{i}=P\left({T}_{i}\le |{t}_{i}|:{\nu }_{i}\right)-P\left({T}_{i}\le -|{t}_{i}|:{\nu }_{i}\right)$.
${\mathbf{tail}}\left(j\right)=\text{'S'}$
The two tail (significance level) probability is returned, i.e., ${p}_{i}=P\left({T}_{i}\ge |{t}_{i}|:{\nu }_{i}\right)+P\left({T}_{i}\le -|{t}_{i}|:{\nu }_{i}\right)$.
Constraint: ${\mathbf{tail}}\left(\mathit{j}\right)=\text{'L'}$, $\text{'U'}$, $\text{'C'}$ or $\text{'S'}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
3: $\mathbf{lt}$Integer Input
On entry: the length of the array t.
Constraint: ${\mathbf{lt}}>0$.
4: $\mathbf{t}\left({\mathbf{lt}}\right)$Real (Kind=nag_wp) array Input
On entry: ${t}_{i}$, the values of the Student's $t$ variates with ${t}_{i}={\mathbf{t}}\left(j\right)$, .
5: $\mathbf{ldf}$Integer Input
On entry: the length of the array df.
Constraint: ${\mathbf{ldf}}>0$.
6: $\mathbf{df}\left({\mathbf{ldf}}\right)$Real (Kind=nag_wp) array Input
On entry: ${\nu }_{i}$, the degrees of freedom of the Student's $t$-distribution with ${\nu }_{i}={\mathbf{df}}\left(j\right)$, .
Constraint: ${\mathbf{df}}\left(\mathit{j}\right)\ge 1.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{ldf}}$.
7: $\mathbf{p}\left(*\right)$Real (Kind=nag_wp) array Output
Note: the dimension of the array p must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lt}},{\mathbf{ldf}}\right)$.
On exit: ${p}_{i}$, the probabilities for the Student's $t$ distribution.
8: $\mathbf{ivalid}\left(*\right)$Integer array Output
Note: the dimension of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lt}},{\mathbf{ldf}}\right)$.
On exit: ${\mathbf{ivalid}}\left(i\right)$ indicates any errors with the input arguments, with
${\mathbf{ivalid}}\left(i\right)=0$
No error.
${\mathbf{ivalid}}\left(i\right)=1$
On entry, invalid value supplied in tail when calculating ${p}_{i}$.
${\mathbf{ivalid}}\left(i\right)=2$
On entry, ${\nu }_{i}<1.0$.
9: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, at least one value of tail or df was invalid.
${\mathbf{ifail}}=2$
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ltail}}>0$.
${\mathbf{ifail}}=3$
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lt}}>0$.
${\mathbf{ifail}}=4$
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldf}}>0$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7Accuracy

The computed probability should be accurate to five significant places for reasonable probabilities but there will be some loss of accuracy for very low probabilities (less than ${10}^{-10}$), see Hastings and Peacock (1975).

8Parallelism and Performance

g01sbf is not threaded in any implementation.

The probabilities could also be obtained by using the appropriate transformation to a beta distribution (see Abramowitz and Stegun (1972)) and using g01sef. This routine allows you to set the required accuracy.

10Example

This example reads values from, and degrees of freedom for Student's $t$-distributions along with the required tail. The probabilities are calculated and printed.

10.1Program Text

Program Text (g01sbfe.f90)

10.2Program Data

Program Data (g01sbfe.d)

10.3Program Results

Program Results (g01sbfe.r)