G01 Chapter Contents
G01 Chapter Introduction
NAG Library Manual

NAG Library Routine DocumentG01SBF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

1  Purpose

G01SBF returns a number of one or two tail probabilities for the Student's $t$-distribution with real degrees of freedom.

2  Specification

 SUBROUTINE G01SBF ( LTAIL, TAIL, LT, T, LDF, DF, P, IVALID, IFAIL)
 INTEGER LTAIL, LT, LDF, IVALID(*), IFAIL REAL (KIND=nag_wp) T(LT), DF(LDF), P(*) CHARACTER(1) TAIL(LTAIL)

3  Description

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:
 $P Ti ≤ ti :νi = Γ νi+1 / 2 πνi Γνi/2 ∫ -∞ ti 1+ Ti2νi -νi+1 / 2 dTi , νi≥1 .$
Computationally, there are two situations:
(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
 $P Ti ≤ ti :νi = 12 + 12 Pβi Bi ≥ νi νi + ti2 :νi/2,12 when ​ ti>0.0 ;$
(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 parameters 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.

4  References

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

5  Parameters

1:     LTAIL – INTEGERInput
On entry: the length of the array TAIL.
Constraint: ${\mathbf{LTAIL}}>0$.
2:     TAIL(LTAIL) – CHARACTER(1) arrayInput
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., ${p}_{i}=P\left({T}_{i}\le \left|{t}_{i}\right|:{\nu }_{i}\right)-\text{}P\left({T}_{i}\le -\left|{t}_{i}\right|:{\nu }_{i}\right)$.
${\mathbf{TAIL}}\left(j\right)=\text{'S'}$
The two tail (significance level) probability is returned, i.e., ${p}_{i}=P\left({T}_{i}\ge \left|{t}_{i}\right|:{\nu }_{i}\right)+\text{}P\left({T}_{i}\le -\left|{t}_{i}\right|:{\nu }_{i}\right)$.
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:     LT – INTEGERInput
On entry: the length of the array T.
Constraint: ${\mathbf{LT}}>0$.
4:     T(LT) – REAL (KIND=nag_wp) arrayInput
On entry: ${t}_{i}$, the values of the Student's $t$ variates with ${t}_{i}={\mathbf{T}}\left(j\right)$, .
5:     LDF – INTEGERInput
On entry: the length of the array DF.
Constraint: ${\mathbf{LDF}}>0$.
6:     DF(LDF) – REAL (KIND=nag_wp) arrayInput
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:     P($*$) – REAL (KIND=nag_wp) arrayOutput
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:     IVALID($*$) – INTEGER arrayOutput
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:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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).

6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, 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: ${\mathbf{LDF}}>0$.
${\mathbf{IFAIL}}=-999$
Dynamic memory allocation failed.

7  Accuracy

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).

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.

9  Example

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.

9.1  Program Text

Program Text (g01sbfe.f90)

9.2  Program Data

Program Data (g01sbfe.d)

9.3  Program Results

Program Results (g01sbfe.r)