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_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.
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)