NAG Library Function Document

nag_deviates_normal (g01fac)

▸▿ 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

nag_deviates_normal (g01fac) returns the deviate associated with the given probability of the standard Normal distribution.

2 Specification

#include <nag.h>

#include <nagg01.h>

double

nag_deviates_normal (Nag_TailProbability tail, double p, NagError *fail)

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: $tail$ – Nag_TailProbabilityInput

On entry: indicates which tail the supplied probability represents.

$tail = Nag_LowerTail$: The lower probability, i.e., $P (X \leq x_{p})$ .
$tail = Nag_UpperTail$: The upper probability, i.e., $P (X \geq x_{p})$ .
$tail = Nag_TwoTailSignif$: The two tail (significance level) probability, i.e., $P (X \geq |x_{p}|) + P (X \leq - |x_{p}|)$ .
$tail = Nag_TwoTailConfid$: The two tail (confidence interval) probability, i.e., $P (X \leq |x_{p}|) - P (X \leq - |x_{p}|)$ .

Constraint:

tail = Nag_LowerTail

Nag_UpperTail

Nag_TwoTailSignif

Nag_TwoTailConfid

2: $p$ – doubleInput

On entry:

p

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

Constraint:

0.0 < p < 1.0

3: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

If on exit $fail . code =$ NE_NOERROR, then nag_deviates_normal (g01fac) returns $0.0$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NE_REAL_ARG_GE: On entry, $p = 〈value〉$ .
Constraint: $p < 1.0$ .
NE_REAL_ARG_LE: On entry, $p = 〈value〉$ .
Constraint: $p > 0.0$ .

7 Accuracy

The accuracy is mainly limited by the machine precision.

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

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

NAG Library Function Documentnag_deviates_normal (g01fac)

▸▿ 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

NAG Library Function Document

nag_deviates_normal (g01fac)