nag_deviates_normal_vector (g01tac) (PDF version)
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g01 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_deviates_normal_vector (g01tac)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_deviates_normal_vector (g01tac) returns a number of deviates associated with given probabilities of the Normal distribution.

2  Specification

#include <nag.h>
#include <nagg01.h>
void  nag_deviates_normal_vector (Integer ltail, const Nag_TailProbability tail[], Integer lp, const double p[], Integer lxmu, const double xmu[], Integer lxstd, const double xstd[], double x[], Integer ivalid[], NagError *fail)

3  Description

The deviate, xpi associated with the lower tail probability, pi, for the Normal distribution is defined as the solution to
PXixpi=pi=-xpiZiXidXi,
where
ZiXi=12πσi2e-Xi-μi2/2σi2, ​-<Xi< .
The method used is an extension of that of Wichura (1988). pi is first replaced by qi=pi-0.5.
(a) If qi0.3, zi is computed by a rational Chebyshev approximation
zi=siAisi2 Bisi2 ,
where si=2πqi and Ai, Bi are polynomials of degree 7.
(b) If 0.3<qi0.42, zi is computed by a rational Chebyshev approximation
zi=signqi Citi Diti ,
where ti=qi-0.3 and Ci, Di are polynomials of degree 5.
(c) If qi>0.42, zi is computed as
zi=signqi Eiui Fiui +ui ,
where ui = -2 × log minpi,1-pi  and Ei, Fi are polynomials of degree 6.
xpi is then calculated from zi, using the relationsship zpi = xi - μi σi .
For the upper tail probability -xpi is returned, while for the two tail probabilities the value xipi* is returned, where pi* is the required tail probability computed from the input value of pi.
The input arrays to this function are designed to allow maximum flexibility in the supply of vector arguments 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.

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:     ltailIntegerInput
On entry: the length of the array tail.
Constraint: ltail>0.
2:     tail[ltail]const Nag_TailProbabilityInput
On entry: indicates which tail the supplied probabilities represent. Letting Z denote a variate from a standard Normal distribution, and zi = xpi - μi σi , then for j= i-1 mod ltail , for i=1,2,,maxltail,lp,lxmu,lxstd:
tail[j]=Nag_LowerTail
The lower tail probability, i.e., pi=PZzi.
tail[j]=Nag_UpperTail
The upper tail probability, i.e., pi=PZzi.
tail[j]=Nag_TwoTailConfid
The two tail (confidence interval) probability, i.e., pi=PZzi-PZ-zi.
tail[j]=Nag_TwoTailSignif
The two tail (significance level) probability, i.e., pi=PZzi+PZ-zi.
Constraint: tail[j-1]=Nag_LowerTail, Nag_UpperTail, Nag_TwoTailConfid or Nag_TwoTailSignif, for j=1,2,,ltail.
3:     lpIntegerInput
On entry: the length of the array p.
Constraint: lp>0.
4:     p[lp]const doubleInput
On entry: pi, the probabilities for the Normal distribution as defined by tail with pi=p[j], j=i-1 mod lp.
Constraint: 0.0<p[j-1]<1.0, for j=1,2,,lp.
5:     lxmuIntegerInput
On entry: the length of the array xmu.
Constraint: lxmu>0.
6:     xmu[lxmu]const doubleInput
On entry: μi, the means with μi=xmu[j], j=i-1 mod lxmu.
7:     lxstdIntegerInput
On entry: the length of the array xstd.
Constraint: lxstd>0.
8:     xstd[lxstd]const doubleInput
On entry: σi, the standard deviations with σi=xstd[j], j=i-1 mod lxstd.
Constraint: xstd[j-1]>0.0, for j=1,2,,lxstd.
9:     x[dim]doubleOutput
Note: the dimension, dim, of the array x must be at least maxltail,lxmu,lxstd,lp.
On exit: xpi, the deviates for the Normal distribution.
10:   ivalid[dim]IntegerOutput
Note: the dimension, dim, of the array ivalid must be at least maxltail,lxmu,lxstd,lp.
On exit: ivalid[i-1] indicates any errors with the input arguments, with
ivalid[i-1]=0
No error.
ivalid[i-1]=1
On entry,invalid value supplied in tail when calculating xpi.
ivalid[i-1]=2
On entry,pi0.0,
orpi1.0.
ivalid[i-1]=3
On entry,σi0.0.
11:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_ARRAY_SIZE
On entry, array size=value.
Constraint: lp>0.
On entry, array size=value.
Constraint: ltail>0.
On entry, array size=value.
Constraint: lxmu>0.
On entry, array size=value.
Constraint: lxstd>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.
NW_IVALID
On entry, at least one value of tail, xstd or p was invalid.
Check ivalid for more information.

7  Accuracy

The accuracy is mainly limited by the machine precision.

8  Further Comments

None.

9  Example

This example reads vectors of values for μi, σi and pi and prints the corresponding deviates.

9.1  Program Text

Program Text (g01tace.c)

9.2  Program Data

Program Data (g01tace.d)

9.3  Program Results

Program Results (g01tace.r)


nag_deviates_normal_vector (g01tac) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012