NAG Library Routine Document

G01SCF

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

G01SCF returns a number of lower or upper tail probabilities for the

χ^{2}

-distribution with real degrees of freedom.

2 Specification

SUBROUTINE G01SCF ( LTAIL, TAIL, LX, X, LDF, DF, P, IVALID, IFAIL)

INTEGER LTAIL, LX, LDF, IVALID(*), IFAIL

REAL (KIND=nag_wp) X(LX), DF(LDF), P(*)

CHARACTER(1) TAIL(LTAIL)

3 Description

The lower tail probability for the

χ^{2}

-distribution with

ν_{i}

degrees of freedom,

P = (X_{i} \leq x_{i} : ν_{i})

is defined by:

P = (X_{i} \leq x_{i} : ν_{i}) = \frac{1}{2^{ν_{i} / 2} Γ (ν_{i} / 2)} \int_{0.0}^{x_{i}} {X_{i}}^{ν_{i} / 2 - 1} e^{- X_{i} / 2} d X_{i}, x_{i} \geq 0, ν_{i} > 0 .

To calculate

P = (X_{i} \leq x_{i} : ν_{i})

a transformation of a gamma distribution is employed, i.e., a

χ^{2}

-distribution with

ν_{i}

degrees of freedom is equal to a gamma distribution with scale parameter

2

and shape parameter

ν_{i} / 2

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 whether the lower or upper tail probabilities are required. For

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

, for

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

$TAIL (j) ='L'$: The lower tail probability is returned, i.e., $p_{i} = P (X_{i} \leq x_{i} : ν_{i})$ .
$TAIL (j) ='U'$: The upper tail probability is returned, i.e., $p_{i} = P (X_{i} \geq x_{i} : ν_{i})$ .

Constraint:

TAIL (j) ='L'

'U'

, 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 values of the

χ^{2}

variates with

ν_{i}

degrees of freedom with

x_{i} = X (j)

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

Constraint:

X (j) \geq 0.0

, for

j = 1, 2, \dots, LX

5: LDF – INTEGERInput

On entry: the length of the array DF.

Constraint:

LDF > 0

6: DF(LDF) – REAL (KIND=nag_wp) arrayInput

On entry:

ν_{i}

, the degrees of freedom of the

χ^{2}

-distribution with

ν_{i} = DF (j)

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

Constraint:

DF (j) > 0.0

, for

j = 1, 2, \dots, LDF

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

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

\max (LTAIL, LDF, LX)

On exit:

p_{i}

, the probabilities for the

χ^{2}

distribution.

8: IVALID( $*$ ) – INTEGER arrayOutput

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

\max (LTAIL, LDF, LX)

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, $x_{i} < 0.0$ .
$IVALID (i) = 3$: On entry, $ν_{i} \leq 0.0$ .
$IVALID (i) = 4$: The solution has failed to converge while calculating the gamma variate. The result returned should represent an approximation to the solution.

9: 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, because for this routine the values of the output parameters may be useful even if

IFAIL \neq 0

on exit, the recommended value is

- 1

. 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).

Note: G01SCF may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, at least one value of X, DF or TAIL was invalid, or the solution failed to converge.
Check IVALID for more information.

$IFAIL = 2$: On entry, $array size = ⟨value⟩$ .
Constraint: $LTAIL > 0$ .

$IFAIL = 3$: On entry, $array size = ⟨value⟩$ .
Constraint: $LX > 0$ .

$IFAIL = 4$: On entry, $array size = ⟨value⟩$ .
Constraint: $LDF > 0$ .

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

7 Accuracy

A relative accuracy of five significant figures is obtained in most cases.

8 Further Comments

For higher accuracy the transformation described in Section 3 may be used with a direct call to S14BAF.

9 Example

Values from various

χ^{2}

-distributions are read, the lower tail probabilities calculated, and all these values printed out.

NAG Library Routine DocumentG01SCF

+− 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

G01SCF