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

P (X_{i} \leq x_{p_{i}}) = p_{i} = \int_{- \infty}^{x_{p_{i}}} Z_{i} (X_{i}) d X_{i},

where

Z_{i} (X_{i}) = \frac{1}{\sqrt{2 π {σ_{i}}^{2}}} e^{- {(X_{i} - μ_{i})}^{2} / (2 {σ_{i}}^{2})}, ​ - \infty < X_{i} < \infty .

The method used is an extension of that of Wichura (1988).

p_{i}

is first replaced by

q_{i} = p_{i} - 0.5

(a)

|q_{i}| \leq 0.3

z_{i}

is computed by a rational Chebyshev approximation

z_{i} = s_{i} \frac{A_{i} ({s_{i}}^{2})}{B_{i} ({s_{i}}^{2})},

where

s_{i} = \sqrt{2 π} q_{i}

and

A_{i}

B_{i}

are polynomials of degree

7

(b)

0.3 < |q_{i}| \leq 0.42

z_{i}

is computed by a rational Chebyshev approximation

z_{i} = sign q_{i} (\frac{C_{i} (t_{i})}{D_{i} (t_{i})}),

where

t_{i} = |q_{i}| - 0.3

and

C_{i}

D_{i}

are polynomials of degree

5

(c)

|q_{i}| > 0.42

z_{i}

is computed as

z_{i} = sign q_{i} [(\frac{E_{i} (u_{i})}{F_{i} (u_{i})}) + u_{i}],

where

u_{i} = \sqrt{- 2 \times \log (\min (p_{i}, 1 - p_{i}))}

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} - μ_{i}}{σ_{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: $ltail$ – IntegerInput

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

z_{i} = \frac{x_{p_{i}} - μ_{i}}{σ_{i}}

, then for

j = (i - 1) mod ltail

, for

i = 1, 2, \dots, \max (ltail, lp, lxmu, lxstd)

$tail [j] = Nag_LowerTail$: The lower tail probability, i.e., $p_{i} = P (Z \leq z_{i})$ .
$tail [j] = Nag_UpperTail$: The upper tail probability, i.e., $p_{i} = P (Z \geq z_{i})$ .
$tail [j] = Nag_TwoTailConfid$: The two tail (confidence interval) probability, i.e., $p_{i} = P (Z \leq |z_{i}|) - P (Z \leq - |z_{i}|)$ .
$tail [j] = Nag_TwoTailSignif$: The two tail (significance level) probability, i.e., $p_{i} = P (Z \geq |z_{i}|) + P (Z \leq - |z_{i}|)$ .

Constraint:

tail [j - 1] = Nag_LowerTail

Nag_UpperTail

Nag_TwoTailConfid

Nag_TwoTailSignif

, for

j = 1, 2, \dots, ltail

3: $lp$ – IntegerInput

On entry: the length of the array p.

Constraint:

lp > 0

4: $p [lp]$ – const doubleInput

On entry:

p_{i}

, the probabilities for the Normal distribution as defined by tail with

p_{i} = p [j]

j = (i - 1) mod lp

Constraint:

0.0 < p [j - 1] < 1.0

, for

j = 1, 2, \dots, lp

5: $lxmu$ – IntegerInput

On entry: the length of the array xmu.

Constraint:

lxmu > 0

6: $xmu [lxmu]$ – const doubleInput

On entry:

μ_{i}

, the means with

μ_{i} = xmu [j]

j = (i - 1) mod lxmu

7: $lxstd$ – IntegerInput

On entry: the length of the array xstd.

Constraint:

lxstd > 0

8: $xstd [lxstd]$ – const doubleInput

On entry:

σ_{i}

, the standard deviations with

σ_{i} = xstd [j]

j = (i - 1) mod lxstd

Constraint:

xstd [j - 1] > 0.0

, for

j = 1, 2, \dots, lxstd

9: $x [\dim]$ – doubleOutput

Note: the dimension, dim, of the array x must be at least

\max (ltail, lxmu, lxstd, lp)

On exit:

x_{p_{i}}

, the deviates for the Normal distribution.

10: $ivalid [\dim]$ – IntegerOutput

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

x_{p_{i}}

$ivalid [i - 1] = 2$

On entry,	$p_{i} \leq 0.0$ ,
or	$p_{i} \geq 1.0$ .

$ivalid [i - 1] = 3$

On entry,

σ_{i} \leq 0.0

11: $fail$ – 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

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_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 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.
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

nag_deviates_normal_vector (g01tac) is not threaded in any implementation.

9

Further Comments

None.

10

Example

This example reads vectors of values for

μ_{i}

σ_{i}

and

p_{i}

and prints the corresponding deviates.

NAG Library Function Document

nag_deviates_normal_vector (g01tac)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results