NAG CL Interface
g01tac (inv_​cdf_​normal_​vector)

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1 Purpose

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

2 Specification

#include <nag.h>
void  g01tac (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)
The function may be called by the names: g01tac, nag_stat_inv_cdf_normal_vector or nag_deviates_normal_vector.

3 Description

The deviate, xpi associated with the lower tail probability, pi, for the Normal distribution is defined as the solution to
P(Xixpi)=pi=-xpiZi(Xi)dXi,  
where
Zi(Xi)=12πσi2e-(Xi-μi)2/(2σi2), ​-<Xi< .  
The method used is an extension of that of Wichura (1988). pi is first replaced by qi=pi-0.5.
  1. (a)If |qi|0.3, zi is computed by a rational Chebyshev approximation
    zi=siAi(si2) Bi(si2) ,  
    where si=2πqi and Ai, Bi are polynomials of degree 7.
  2. (b)If 0.3<|qi|0.42, zi is computed by a rational Chebyshev approximation
    zi=signqi (Ci(ti) Di(ti) ) ,  
    where ti=|qi|-0.3 and Ci, Di are polynomials of degree 5.
  3. (c)If |qi|>0.42, zi is computed as
    zi=signqi [(Ei(ui) Fi(ui) )+ui] ,  
    where ui = −2 × log(min(pi,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

NIST Digital Library of Mathematical Functions
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: ltail Integer Input
On entry: the length of the array tail.
Constraint: ltail>0.
2: tail[ltail] const Nag_TailProbability Input
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,,max(ltail,lp,lxmu,lxstd):
tail[j]=Nag_LowerTail
The lower tail probability, i.e., pi=P(Zzi).
tail[j]=Nag_UpperTail
The upper tail probability, i.e., pi=P(Zzi).
tail[j]=Nag_TwoTailConfid
The two tail (confidence interval) probability, i.e., pi=P(Z|zi|)-P(Z-|zi|).
tail[j]=Nag_TwoTailSignif
The two tail (significance level) probability, i.e., pi=P(Z|zi|)+P(Z-|zi|).
Constraint: tail[j-1]=Nag_LowerTail, Nag_UpperTail, Nag_TwoTailConfid or Nag_TwoTailSignif, for j=1,2,,ltail.
3: lp Integer Input
On entry: the length of the array p.
Constraint: lp>0.
4: p[lp] const double Input
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: lxmu Integer Input
On entry: the length of the array xmu.
Constraint: lxmu>0.
6: xmu[lxmu] const double Input
On entry: μi, the means with μi=xmu[j], j=(i-1) mod lxmu.
7: lxstd Integer Input
On entry: the length of the array xstd.
Constraint: lxstd>0.
8: xstd[lxstd] const double Input
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] double Output
Note: the dimension, dim, of the array x must be at least max(ltail,lxmu,lxstd,lp).
On exit: xpi, the deviates for the Normal distribution.
10: ivalid[dim] Integer Output
Note: the dimension, dim, of the array ivalid must be at least max(ltail,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, or, pi1.0.
ivalid[i-1]=3
On entry, σi0.0.
11: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
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 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g01tac is not threaded in any implementation.

9 Further Comments

None.

10 Example

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

10.1 Program Text

Program Text (g01tace.c)

10.2 Program Data

Program Data (g01tace.d)

10.3 Program Results

Program Results (g01tace.r)