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,

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)If $| q_{i} | \leq 0.3$ , $z_{i}$ 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)If $0.3 < | q_{i} | \leq 0.42$ , $z_{i}$ 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)If $| q_{i} | > 0.42$ , $z_{i}$ 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

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

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: $lp$ – Integer Input

On entry: the length of the array p.

Constraint:

lp > 0

4: $p [lp]$ – const double Input

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: $lxmu$ – Integer Input

On entry: the length of the array xmu.

Constraint:

lxmu > 0

6: $xmu [lxmu]$ – const double Input

On entry:

μ_{i}

, the means with

μ_{i} = 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:

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: $x [\dim]$ – double Output

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: $ivalid [\dim]$ – 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 $x_{p_{i}}$ .
$ivalid [i - 1] = 2$: 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 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

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.

g01ta: FL CL CPP AD

NAG CL Interfaceg01tac (inv_​cdf_​normal_​vector)

▸▿ 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

NAG CL Interface
g01tac (inv_cdf_normal_vector)