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NAG Library Routine Document

g01faf (inv_cdf_normal)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

g01faf returns the deviate associated with the given probability of the standard Normal distribution, via the routine name.

2

Specification

Fortran Interface

Function g01faf (

tail, p, ifail)

Real (Kind=nag_wp)	::	g01faf
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	p
Character (1), Intent (In)	::	tail

C Header Interface

#include nagmk26.h

double

g01faf_ ( const char *tail, const double *p, Integer *ifail, const Charlen length_tail)

3

Description

The deviate,

x_{p}

associated with the lower tail probability,

p

, for the standard Normal distribution is defined as the solution to

P (X \leq x_{p}) = p = \int_{- \infty}^{x_{p}} Z (X) d X,

where

Z (X) = \frac{1}{\sqrt{2 π}} e^{- X^{2} / 2}, - \infty < X < \infty .

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

p

is first replaced by

q = p - 0.5

(a)

|q| \leq 0.3

x_{p}

is computed by a rational Chebyshev approximation

x_{p} = s \frac{A (s^{2})}{B (s^{2})},

where

s = \sqrt{2 π} q

and

A

B

are polynomials of degree

7

(b)

0.3 < |q| \leq 0.42

x_{p}

is computed by a rational Chebyshev approximation

x_{p} = sign q (\frac{C (t)}{D (t)}),

where

t = |q| - 0.3

and

C

D

are polynomials of degree

5

(c)

|q| > 0.42

x_{p}

is computed as

x_{p} = sign q [(\frac{E (u)}{F (u)}) + u],

where

u = \sqrt{- 2 \times \log (\min (p, 1 - p))}

and

E

F

are polynomials of degree

6

For the upper tail probability

- x_{p}

is returned, while for the two tail probabilities the value

x_{p^{*}}

is returned, where

p^{*}

is the required tail probability computed from the input value of

p

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

Wichura (1988) Algorithm AS 241: the percentage points of the Normal distribution Appl. Statist. 37 477–484

5

Arguments

1: $tail$ – Character(1)Input

On entry: indicates which tail the supplied probability represents.

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

Constraint:

tail ='L'

'U'

'S'

'C'

2: $p$ – Real (Kind=nag_wp)Input

On entry:

p

, the probability from the standard Normal distribution as defined by tail.

Constraint:

0.0 < p < 1.0

3: $ifail$ – IntegerInput/Output

On entry: ifail must be set to

0

- 1 ​ 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 ​ 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 $- 1 or 1$ 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:

If on exit

ifail \neq 0

, then g01faf returns

0.0

$ifail = 1$: On entry, $tail = 〈value〉$ .
Constraint: $tail ='L'$ , $'U'$ , $'S'$ or $'C'$ .

$ifail = 2$: On entry, $p = 〈value〉$ .
Constraint: $p < 1.0$ .

On entry, $p = 〈value〉$ .
Constraint: $p > 0.0$ .

$ifail = - 99$: 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.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

The accuracy is mainly limited by the machine precision.

8

Parallelism and Performance

g01faf is not threaded in any implementation.

9

Further Comments

None.

10

Example

Four values of tail and p are input and the deviates calculated and printed.

NAG Library Routine Document

g01faf (inv_cdf_normal)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results