NAG Library Routine Document

G01SAF

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G01SAF returns a number of one or two tail probabilities for the Normal distribution.

2 Specification

SUBROUTINE G01SAF ( LTAIL, TAIL, LX, X, LXMU, XMU, LXSTD, XSTD, P, IVALID, IFAIL)

INTEGER LTAIL, LX, LXMU, LXSTD, IVALID(*), IFAIL

REAL (KIND=nag_wp) X(LX), XMU(LXMU), XSTD(LXSTD), P(*)

CHARACTER(1) TAIL(LTAIL)

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 S15ADF.

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 S15AEF.

The input arrays to this routine are designed to allow maximum flexibility in the supply of vector parameters 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

5 Parameters

1: LTAIL – INTEGERInput

On entry: the length of the array TAIL.

Constraint:

LTAIL > 0

2: TAIL(LTAIL) – CHARACTER(1) arrayInput

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) + 1

, for

i = 1, 2, \dots, \max (LX, LTAIL, LXMU, LXSTD)

$TAIL (j) ='L'$: The lower tail probability is returned, i.e., $p_{i} = P (Z \leq z_{i})$ .
$TAIL (j) ='U'$: The upper tail probability is returned, i.e., $p_{i} = P (Z \geq z_{i})$ .
$TAIL (j) ='C'$: The two tail (confidence interval) probability is returned, i.e., $p_{i} = P (Z \leq |z_{i}|) - P (Z \leq - |z_{i}|)$ .
$TAIL (j) ='S'$: 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) ='L'

'U'

'C'

'S'

, for

j = 1, 2, \dots, LTAIL

3: LX – INTEGERInput

On entry: the length of the array X.

Constraint:

LX > 0

4: X(LX) – REAL (KIND=nag_wp) arrayInput

On entry:

x_{i}

, the Normal variate values with

x_{i} = X (j)

j = ((i - 1) mod LX) + 1

5: LXMU – INTEGERInput

On entry: the length of the array XMU.

Constraint:

LXMU > 0

6: XMU(LXMU) – REAL (KIND=nag_wp) arrayInput

On entry:

μ_{i}

, the means with

μ_{i} = XMU (j)

j = ((i - 1) mod LXMU) + 1

7: LXSTD – INTEGERInput

On entry: the length of the array XSTD.

Constraint:

LXSTD > 0

8: XSTD(LXSTD) – REAL (KIND=nag_wp) arrayInput

On entry:

σ_{i}

, the standard deviations with

σ_{i} = XSTD (j)

j = ((i - 1) mod LXSTD) + 1

Constraint:

XSTD (j) > 0.0

, for

j = 1, 2, \dots, LXSTD

9: P( $*$ ) – REAL (KIND=nag_wp) arrayOutput

Note: the dimension 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( $*$ ) – INTEGER arrayOutput

Note: the dimension of the array IVALID must be at least

\max (LX, LTAIL, LXMU, LXSTD)

On exit:

IVALID (i)

indicates any errors with the input arguments, with

$IVALID (i) = 0$: No error.
$IVALID (i) = 1$: On entry, invalid value supplied in TAIL when calculating $p_{i}$ .
$IVALID (i) = 2$: On entry, $σ_{i} \leq 0.0$ .

11: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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

- 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: $LXSTD > 0$ .

$IFAIL = - 999$: Dynamic memory allocation failed.

7 Accuracy

Accuracy is limited by machine precision. For detailed error analysis see S15ADF and S15AEF.

8 Further Comments

None.

9 Example

Four values of TAIL, X, XMU and XSTD are input and the probabilities calculated and printed.

NAG Library Routine DocumentG01SAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G01SAF