G01TAF : NAG Library, Mark 24

3 Description

The deviate,

x_{p_{i}}

associated with the lower tail probability,

p_{i}

, for the Normal distribution is defined as the solution to

P (X_{i} \leq x_{p_{i}}) = p_{i} = \int_{- \infty}^{x_{p_{i}}} Z_{i} (X_{i}) d X_{i},

where

Z_{i} (X_{i}) = \frac{1}{\sqrt{2 π {σ_{i}}^{2}}} e^{- {(X_{i} - μ_{i})}^{2} / (2 {σ_{i}}^{2})}, ​ - \infty < X_{i} < \infty .

The method used is an extension of that of Wichura (1988).

p_{i}

is first replaced by

q_{i} = p_{i} - 0.5

(a)

|q_{i}| \leq 0.3

z_{i}

is computed by a rational Chebyshev approximation

z_{i} = s_{i} \frac{A_{i} ({s_{i}}^{2})}{B_{i} ({s_{i}}^{2})},

where

s_{i} = \sqrt{2 π} q_{i}

and

A_{i}

B_{i}

are polynomials of degree

7

(b)

0.3 < |q_{i}| \leq 0.42

z_{i}

is computed by a rational Chebyshev approximation

z_{i} = sign q_{i} (\frac{C_{i} (t_{i})}{D_{i} (t_{i})}),

where

t_{i} = |q_{i}| - 0.3

and

C_{i}

D_{i}

are polynomials of degree

5

(c)

|q_{i}| > 0.42

z_{i}

is computed as

z_{i} = sign q_{i} [(\frac{E_{i} (u_{i})}{F_{i} (u_{i})}) + u_{i}],

where

u_{i} = \sqrt{- 2 \times \log (\min (p_{i}, 1 - p_{i}))}

and

E_{i}

F_{i}

are polynomials of degree

6

x_{p_{i}}

is then calculated from

z_{i}

, using the relationsship

z_{p_{i}} = \frac{x_{i} - μ_{i}}{σ_{i}}

For the upper tail probability

- x_{p_{i}}

is returned, while for the two tail probabilities the value

x_{i {p_{i}}^{*}}

is returned, where

{p_{i}}^{*}

is the required tail probability computed from the input value of

p_{i}

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.

5 Parameters

1: LTAIL – INTEGERInput

On entry: the length of the array TAIL.

Constraint:

LTAIL > 0

2: TAIL(LTAIL) – CHARACTER(1) arrayInput

On entry: indicates which tail the supplied probabilities represent. Letting

Z

denote a variate from a standard Normal distribution, and

z_{i} = \frac{x_{p_{i}} - μ_{i}}{σ_{i}}

, then for

j = ((i - 1) mod LTAIL) + 1

, for

i = 1, 2, \dots, \max (LTAIL, LP, LXMU, LXSTD)

$TAIL (j) ='L'$: The lower tail probability, i.e., $p_{i} = P (Z \leq z_{i})$ .
$TAIL (j) ='U'$: The upper tail probability, i.e., $p_{i} = P (Z \geq z_{i})$ .
$TAIL (j) ='C'$: The two tail (confidence interval) probability, i.e., $p_{i} = P (Z \leq |z_{i}|) - P (Z \leq - |z_{i}|)$ .
$TAIL (j) ='S'$: The two tail (significance level) probability, i.e., $p_{i} = P (Z \geq |z_{i}|) + P (Z \leq - |z_{i}|)$ .

Constraint:

TAIL (j) ='L'

'U'

'C'

'S'

, for

j = 1, 2, \dots, LTAIL

3: LP – INTEGERInput

On entry: the length of the array P.

Constraint:

LP > 0

4: P(LP) – REAL (KIND=nag_wp) arrayInput

On entry:

p_{i}

, the probabilities for the Normal distribution as defined by TAIL with

p_{i} = P (j)

j = (i - 1) mod LP + 1

Constraint:

0.0 < P (j) < 1.0

, for

j = 1, 2, \dots, LP

5: LXMU – INTEGERInput

On entry: the length of the array XMU.

Constraint:

LXMU > 0

6: XMU(LXMU) – REAL (KIND=nag_wp) arrayInput

On entry:

μ_{i}

, the means with

μ_{i} = XMU (j)

j = ((i - 1) mod LXMU) + 1

7: LXSTD – INTEGERInput

On entry: the length of the array XSTD.

Constraint:

LXSTD > 0

8: XSTD(LXSTD) – REAL (KIND=nag_wp) arrayInput

On entry:

σ_{i}

, the standard deviations with

σ_{i} = XSTD (j)

j = ((i - 1) mod LXSTD) + 1

Constraint:

XSTD (j) > 0.0

, for

j = 1, 2, \dots, LXSTD

9: X(

*

) – REAL (KIND=nag_wp) arrayOutput

Note: the dimension of the array X must be at least

\max (LTAIL, LXMU, LXSTD, LP)

On exit:

x_{p_{i}}

, the deviates for the Normal distribution.

10: IVALID(

*

) – INTEGER arrayOutput

Note: the dimension of the array IVALID must be at least

\max (LTAIL, LXMU, LXSTD, LP)

On exit:

IVALID (i)

indicates any errors with the input arguments, with

$IVALID (i) = 0$

No error.

$IVALID (i) = 1$

On entry,

invalid value supplied in TAIL when calculating

x_{p_{i}}

$IVALID (i) = 2$

On entry,	$p_{i} \leq 0.0$ ,
or	$p_{i} \geq 1.0$ .

$IVALID (i) = 3$

On entry,

σ_{i} \leq 0.0

11: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ 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 ​ 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 $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

IFAIL = 1

On entry, at least one value of TAIL, XSTD or P was invalid.
Check IVALID for more information.

IFAIL = 2

On entry,

array size = ⟨value⟩

.
Constraint:

LTAIL > 0

IFAIL = 3

On entry,

array size = ⟨value⟩

.
Constraint:

LP > 0

IFAIL = 4

On entry,

array size = ⟨value⟩

.
Constraint:

LXMU > 0

IFAIL = 5

On entry,

array size = ⟨value⟩

.
Constraint:

LXSTD > 0

IFAIL = - 999

Dynamic memory allocation failed.

NAG Library Routine Document

G01TAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine DocumentG01TAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G01TAF