NAG Library Routine Document

g01saf (prob_normal_vector)

1
Purpose

g01saf returns a number of one or two tail probabilities for the Normal distribution.

2
Specification

Fortran Interface
Subroutine g01saf ( ltail, tail, lx, x, lxmu, xmu, lxstd, xstd, p, ivalid, ifail)
Integer, Intent (In):: ltail, lx, lxmu, lxstd
Integer, Intent (Inout):: ifail
Integer, Intent (Out):: ivalid(*)
Real (Kind=nag_wp), Intent (In):: x(lx), xmu(lxmu), xstd(lxstd)
Real (Kind=nag_wp), Intent (Out):: p(*)
Character (1), Intent (In):: tail(ltail)
C Header Interface
#include <nagmk26.h>
void  g01saf_ (const Integer *ltail, const char tail[], const Integer *lx, const double x[], const Integer *lxmu, const double xmu[], const Integer *lxstd, const double xstd[], double p[], Integer ivalid[], Integer *ifail, const Charlen length_tail)

3
Description

The lower tail probability for the Normal distribution, PXixi is defined by:
PXixi = - xi ZiXidXi ,  
where
ZiXi = 1 2πσi2 e -Xi-μi2/2σi2 , - < Xi < .  
The relationship
P Xi xi = 12 erfc - xi - μi 2 σi  
is used, where erfc is the complementary error function, and is computed using s15adf.
When the two tail confidence probability is required the relationship
P Xixi - P Xi - xi = erf xi - μi 2 σi ,  
is used, where erf is the error function, and is computed using s15aef.
The input arrays to this routine 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

5
Arguments

1:     ltail – IntegerInput
On entry: the length of the array tail.
Constraint: ltail>0.
2:     tailltail – Character(1) arrayInput
On entry: indicates which tail the returned probabilities should represent. Letting Z denote a variate from a standard Normal distribution, and zi = xi - μi σi , then for j= i-1 mod ltail +1 , for i=1,2,,maxlx,ltail,lxmu,lxstd:
tailj='L'
The lower tail probability is returned, i.e., pi=PZzi.
tailj='U'
The upper tail probability is returned, i.e., pi=PZzi.
tailj='C'
The two tail (confidence interval) probability is returned, i.e., pi=PZzi-PZ-zi.
tailj='S'
The two tail (significance level) probability is returned, i.e., pi=PZzi+PZ-zi.
Constraint: tailj='L', 'U', 'C' or 'S', for j=1,2,,ltail.
3:     lx – IntegerInput
On entry: the length of the array x.
Constraint: lx>0.
4:     xlx – Real (Kind=nag_wp) arrayInput
On entry: xi, the Normal variate values with xi=xj, j=i-1 mod lx+1.
5:     lxmu – IntegerInput
On entry: the length of the array xmu.
Constraint: lxmu>0.
6:     xmulxmu – Real (Kind=nag_wp) arrayInput
On entry: μi, the means with μi=xmuj, j=i-1 mod lxmu+1.
7:     lxstd – IntegerInput
On entry: the length of the array xstd.
Constraint: lxstd>0.
8:     xstdlxstd – Real (Kind=nag_wp) arrayInput
On entry: σi, the standard deviations with σi=xstdj, j=i-1 mod lxstd+1.
Constraint: xstdj>0.0, for j=1,2,,lxstd.
9:     p* – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array p must be at least maxlx,ltail,lxmu,lxstd.
On exit: pi, the probabilities for the Normal distribution.
10:   ivalid* – Integer arrayOutput
Note: the dimension of the array ivalid must be at least maxlx,ltail,lxmu,lxstd.
On exit: ivalidi indicates any errors with the input arguments, with
ivalidi=0
No error.
ivalidi=1
On entry,invalid value supplied in tail when calculating pi.
ivalidi=2
On entry,σi0.0.
11:   ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1 or 1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 or 1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6
Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
ifail=1
On entry, at least one value of tail or xstd was invalid.
Check ivalid for more information.
ifail=2
On entry, ltail=value.
Constraint: ltail>0.
ifail=3
On entry, lx=value.
Constraint: lx>0.
ifail=4
On entry, lxmu=value.
Constraint: lxmu>0.
ifail=5
On entry, lxstd=value.
Constraint: lxstd>0.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7
Accuracy

Accuracy is limited by machine precision. For detailed error analysis see s15adf and s15aef.

8
Parallelism and Performance

g01saf is not threaded in any implementation.

9
Further Comments

None.

10
Example

Four values of tail, x, xmu and xstd are input and the probabilities calculated and printed.

10.1
Program Text

Program Text (g01safe.f90)

10.2
Program Data

Program Data (g01safe.d)

10.3
Program Results

Program Results (g01safe.r)