void	g01sac (Integer ltail, const Nag_TailProbability tail[], Integer lx, const double x[], Integer lxmu, const double xmu[], Integer lxstd, const double xstd[], double p[], Integer ivalid[], NagError *fail)

The function may be called by the names: g01sac, nag_stat_prob_normal_vector or nag_prob_normal_vector.

3 Description

The lower tail probability for the Normal distribution,

P (X_{i} \leq x_{i})

is defined by:

P (X_{i} \leq x_{i}) = \int_{- \infty}^{x_{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 relationship

P (X_{i} \leq x_{i}) = \frac{1}{2} erfc (\frac{- (x_{i} - μ_{i})}{\sqrt{2} σ_{i}})

is used, where erfc is the complementary error function, and is computed using s15adc.

When the two tail confidence probability is required the relationship

P (X_{i} \leq | x_{i} |) - P (X_{i} \leq - | x_{i} |) = \erf (\frac{| x_{i} - μ_{i} |}{\sqrt{2} σ_{i}}),

is used, where erf is the error function, and is computed using s15aec.

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

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 returned probabilities should represent. Letting

Z

denote a variate from a standard Normal distribution, and

z_{i} = \frac{x_{i} - μ_{i}}{σ_{i}}

, then for

j = (i - 1) mod ltail

, for

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

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

On entry: the length of the array x.

Constraint:

lx > 0

4: $x [lx]$ – const double Input

On entry:

x_{i}

, the Normal variate values with

x_{i} = x [j]

j = (i - 1) mod lx

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: $p [\dim]$ – double Output

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

\max (lx, ltail, lxmu, lxstd)

On exit:

p_{i}

, the probabilities for the Normal distribution.

10: $ivalid [\dim]$ – Integer Output

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

\max (lx, ltail, lxmu, lxstd)

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 $p_{i}$ .
$ivalid [i - 1] = 2$: 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, $ltail = ⟨ value ⟩$ .
Constraint: $ltail > 0$ .

On entry, $lx = ⟨ value ⟩$ .
Constraint: $lx > 0$ .

On entry, $lxmu = ⟨ value ⟩$ .
Constraint: $lxmu > 0$ .

On entry, $lxstd = ⟨ value ⟩$ .
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 or xstd was invalid.
Check ivalid for more information.

7 Accuracy

Accuracy is limited by machine precision. For detailed error analysis see s15adc and s15aec.

8 Parallelism and Performance

g01sac 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.

g01sa: FL CL CPP AD

NAG CL Interfaceg01sac (prob_​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
g01sac (prob_normal_vector)