g01 Chapter Contents
g01 Chapter Introduction
NAG Library Manual

NAG Library Function Documentnag_deviates_normal_vector (g01tac)

1  Purpose

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

2  Specification

 #include #include
 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, ${x}_{{p}_{i}}$ associated with the lower tail probability, ${p}_{i}$, for the Normal distribution is defined as the solution to
 $PXi≤xpi=pi=∫-∞xpiZiXidXi,$
where
 $ZiXi=12πσi2e-Xi-μi2/2σi2, ​-∞
The method used is an extension of that of Wichura (1988). ${p}_{i}$ is first replaced by ${q}_{i}={p}_{i}-0.5$.
(a) If $\left|{q}_{i}\right|\le 0.3$, ${z}_{i}$ is computed by a rational Chebyshev approximation
 $zi=siAisi2 Bisi2 ,$
where ${s}_{i}=\sqrt{2\pi }{q}_{i}$ and ${A}_{i}$, ${B}_{i}$ are polynomials of degree $7$.
(b) If $0.3<\left|{q}_{i}\right|\le 0.42$, ${z}_{i}$ is computed by a rational Chebyshev approximation
 $zi=sign⁡qi Citi Diti ,$
where ${t}_{i}=\left|{q}_{i}\right|-0.3$ and ${C}_{i}$, ${D}_{i}$ are polynomials of degree $5$.
(c) If $\left|{q}_{i}\right|>0.42$, ${z}_{i}$ is computed as
 $zi=sign⁡qi Eiui Fiui +ui ,$
where ${u}_{i}=\sqrt{-2×\mathrm{log}\left(\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({p}_{i},1-{p}_{i}\right)\right)}$ and ${E}_{i}$, ${F}_{i}$ are polynomials of degree $6$.
${x}_{{p}_{i}}$ is then calculated from ${z}_{i}$, using the relationsship ${z}_{{p}_{i}}=\frac{{x}_{i}-{\mu }_{i}}{{\sigma }_{i}}$.
For the upper tail probability $-{x}_{{p}_{i}}$ is returned, while for the two tail probabilities the value ${x}_{i{{p}_{i}}^{*}}$ is returned, where ${{p}_{i}}^{*}$ is the required tail probability computed from the input value of ${p}_{i}$.
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:    $\mathbf{ltail}$IntegerInput
On entry: the length of the array tail.
Constraint: ${\mathbf{ltail}}>0$.
2:    $\mathbf{tail}\left[{\mathbf{ltail}}\right]$const Nag_TailProbabilityInput
On entry: indicates which tail the supplied probabilities represent. Letting $Z$ denote a variate from a standard Normal distribution, and ${z}_{i}=\frac{{x}_{{p}_{i}}-{\mu }_{i}}{{\sigma }_{i}}$, then for , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{lxmu}},{\mathbf{lxstd}}\right)$:
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_LowerTail}$
The lower tail probability, i.e., ${p}_{i}=P\left(Z\le {z}_{i}\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_UpperTail}$
The upper tail probability, i.e., ${p}_{i}=P\left(Z\ge {z}_{i}\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_TwoTailConfid}$
The two tail (confidence interval) probability, i.e., ${p}_{i}=P\left(Z\le \left|{z}_{i}\right|\right)-P\left(Z\le -\left|{z}_{i}\right|\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_TwoTailSignif}$
The two tail (significance level) probability, i.e., ${p}_{i}=P\left(Z\ge \left|{z}_{i}\right|\right)+P\left(Z\le -\left|{z}_{i}\right|\right)$.
Constraint: ${\mathbf{tail}}\left[\mathit{j}-1\right]=\mathrm{Nag_LowerTail}$, $\mathrm{Nag_UpperTail}$, $\mathrm{Nag_TwoTailConfid}$ or $\mathrm{Nag_TwoTailSignif}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
3:    $\mathbf{lp}$IntegerInput
On entry: the length of the array p.
Constraint: ${\mathbf{lp}}>0$.
4:    $\mathbf{p}\left[{\mathbf{lp}}\right]$const doubleInput
On entry: ${p}_{i}$, the probabilities for the Normal distribution as defined by tail with ${p}_{i}={\mathbf{p}}\left[j\right]$, .
Constraint: $0.0<{\mathbf{p}}\left[\mathit{j}-1\right]<1.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lp}}$.
5:    $\mathbf{lxmu}$IntegerInput
On entry: the length of the array xmu.
Constraint: ${\mathbf{lxmu}}>0$.
6:    $\mathbf{xmu}\left[{\mathbf{lxmu}}\right]$const doubleInput
On entry: ${\mu }_{i}$, the means with ${\mu }_{i}={\mathbf{xmu}}\left[j\right]$, .
7:    $\mathbf{lxstd}$IntegerInput
On entry: the length of the array xstd.
Constraint: ${\mathbf{lxstd}}>0$.
8:    $\mathbf{xstd}\left[{\mathbf{lxstd}}\right]$const doubleInput
On entry: ${\sigma }_{i}$, the standard deviations with ${\sigma }_{i}={\mathbf{xstd}}\left[j\right]$, .
Constraint: ${\mathbf{xstd}}\left[\mathit{j}-1\right]>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lxstd}}$.
9:    $\mathbf{x}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lxmu}},{\mathbf{lxstd}},{\mathbf{lp}}\right)$.
On exit: ${x}_{{p}_{i}}$, the deviates for the Normal distribution.
10:  $\mathbf{ivalid}\left[\mathit{dim}\right]$IntegerOutput
Note: the dimension, dim, of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lxmu}},{\mathbf{lxstd}},{\mathbf{lp}}\right)$.
On exit: ${\mathbf{ivalid}}\left[i-1\right]$ indicates any errors with the input arguments, with
${\mathbf{ivalid}}\left[i-1\right]=0$
No error.
${\mathbf{ivalid}}\left[i-1\right]=1$
 On entry, invalid value supplied in tail when calculating ${x}_{{p}_{i}}$.
${\mathbf{ivalid}}\left[i-1\right]=2$
 On entry, ${p}_{i}\le 0.0$, or ${p}_{i}\ge 1.0$.
${\mathbf{ivalid}}\left[i-1\right]=3$
 On entry, ${\sigma }_{i}\le 0.0$.
11:  $\mathbf{fail}$NagError *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.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_ARRAY_SIZE
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lp}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ltail}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lxmu}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lxstd}}>0$.
On entry, argument $〈\mathit{\text{value}}〉$ 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 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NW_IVALID
On entry, at least one value of tail, xstd or p was invalid.

7  Accuracy

The accuracy is mainly limited by the machine precision.

Not applicable.

None.

10  Example

This example reads vectors of values for ${\mu }_{i}$, ${\sigma }_{i}$ and ${p}_{i}$ 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)