NAG Library Routine Document

g01tbf  (inv_cdf_students_t_vector)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{ltail}$ – IntegerInput

On entry: the length of the array tail.

Constraint: $\mathbf{ltail}>0$.

2:     $\mathbf{tail}\left(\mathbf{ltail}\right)$ – Character(1) arrayInput

On entry: indicates which tail the supplied probabilities represent. For $j=\left(\left(\mathit{i}-1\right)\mathrm{ mod }\mathbf{ltail}\right)+1$, for $\mathit{i}=1,2,\dots ,\max \left(\mathbf{ltail},\mathbf{lp},\mathbf{ldf}\right)$:

$\mathbf{tail}\left(j\right)=\text{'L'}$

The lower tail probability, i.e., $p_i=P\left(T_i\le t_{p_i}:{\nu }_i\right)$.

$\mathbf{tail}\left(j\right)=\text{'U'}$

The upper tail probability, i.e., $p_i=P\left(T_i\ge t_{p_i}:{\nu }_i\right)$.

$\mathbf{tail}\left(j\right)=\text{'C'}$

The two tail (confidence interval) probability,
i.e., $p_i=P\left(T_i\le \left|t_{p_i}\right|:{\nu }_i\right)-P\left(T_i\le -\left|t_{p_i}\right|:{\nu }_i\right)$.

$\mathbf{tail}\left(j\right)=\text{'S'}$

The two tail (significance level) probability,
i.e., $p_i=P\left(T_i\ge \left|t_{p_i}\right|:{\nu }_i\right)+P\left(T_i\le -\left|t_{p_i}\right|:{\nu }_i\right)$.

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:     $\mathbf{lp}$ – IntegerInput

On entry: the length of the array p.

Constraint: $\mathbf{lp}>0$.

4:     $\mathbf{p}\left(\mathbf{lp}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: $p_i$, the probability of the required Student's $t$-distribution as defined by tail with $p_i=\mathbf{p}\left(j\right)$, $j=\left(\left(i-1\right)\mathrm{ mod }\mathbf{lp}\right)+1$.

Constraint: $0.0<\mathbf{p}\left(\mathit{j}\right)<1.0$, for $\mathit{j}=1,2,\dots ,\mathbf{lp}$.

5:     $\mathbf{ldf}$ – IntegerInput

On entry: the length of the array df.

Constraint: $\mathbf{ldf}>0$.

6:     $\mathbf{df}\left(\mathbf{ldf}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: ${\nu }_i$, the degrees of freedom of the Student's $t$-distribution with ${\nu }_i=\mathbf{df}\left(j\right)$, $j=\left(\left(i-1\right)\mathrm{ mod }\mathbf{ldf}\right)+1$.

Constraint: $\mathbf{df}\left(\mathit{j}\right)\ge 1.0$, for $\mathit{j}=1,2,\dots ,\mathbf{ldf}$.

7:     $\mathbf{t}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput

Note: the dimension of the array t must be at least $\max \left(\mathbf{ltail},\mathbf{lp},\mathbf{ldf}\right)$.

On exit: $t_{p_i}$, the deviates for the Student's $t$-distribution.

8:     $\mathbf{ivalid}\left(*\right)$ – Integer arrayOutput

Note: the dimension of the array ivalid must be at least $\max \left(\mathbf{ltail},\mathbf{lp},\mathbf{ldf}\right)$.

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 $t_{p_i}$.

$\mathbf{ivalid}\left(i\right)=2$

On entry,	$p_i\le 0.0$,
or	$p_i\ge 1.0$.

$\mathbf{ivalid}\left(i\right)=3$

On entry,

${\nu }_i<1.0$.

$\mathbf{ivalid}\left(i\right)=4$

The solution has failed to converge. The result returned should represent an approximation to the solution.

9:     $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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

$\mathbf{ifail}=1$

On entry, at least one value of tail, p or df was invalid, or the solution failed to converge.
Check ivalid for more information.

$\mathbf{ifail}=2$

On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ltail}>0$.

$\mathbf{ifail}=3$

On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{lp}>0$.

$\mathbf{ifail}=4$

On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ldf}>0$.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (g01tbfe.f90)

10.2

Program Data

Program Data (g01tbfe.d)

10.3

Program Results

Program Results (g01tbfe.r)