NAG Library Function Document

nag_deviates_normal (g01fac)

 Contents

    1  Purpose
    7  Accuracy

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, xp associated with the lower tail probability, p, for the standard Normal distribution is defined as the solution to
PXxp=p=-xpZXdX,  
where
ZX=12πe-X2/2,   -<X< .  
The method used is an extension of that of Wichura (1988). p is first replaced by q=p-0.5.
(a) If q0.3, xp is computed by a rational Chebyshev approximation
xp=sAs2 Bs2 ,  
where s=2πq and A, B are polynomials of degree 7.
(b) If 0.3<q0.42, xp is computed by a rational Chebyshev approximation
xp=signq Ct Dt ,  
where t=q-0.3 and C, D are polynomials of degree 5.
(c) If q>0.42, xp is computed as
xp=signq Eu Fu +u ,  
where u = -2 × log minp,1-p  and E, F are polynomials of degree 6.
For the upper tail probability -xp is returned, while for the two tail probabilities the value xp* 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., PXxp.
tail=Nag_UpperTail
The upper probability, i.e., PXxp.
tail=Nag_TwoTailSignif
The two tail (significance level) probability, i.e., PXxp+PX-xp.
tail=Nag_TwoTailConfid
The two tail (confidence interval) probability, i.e., PXxp-PX-xp.
Constraint: tail=Nag_LowerTail, Nag_UpperTail, Nag_TwoTailSignif or 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 3.7 in How to Use the NAG Library and its Documentation).

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 2.3.1.2 in How to Use the NAG Library and its Documentation 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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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

nag_deviates_normal (g01fac) 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.

10.1
Program Text

Program Text (g01face.c)

10.2
Program Data

Program Data (g01face.d)

10.3
Program Results

Program Results (g01face.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017