NAG Library Routine Document

G01TCF

+− 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

G01TCF returns a number of deviates associated with the given probabilities of the

χ^{2}

-distribution with real degrees of freedom.

2 Specification

SUBROUTINE G01TCF (

LTAIL, TAIL, LP, P, LDF, DF, X, IVALID, IFAIL)

INTEGER	LTAIL, LP, LDF, IVALID(*), IFAIL
REAL (KIND=nag_wp)	P(LP), DF(LDF), X(*)
CHARACTER(1)	TAIL(LTAIL)

3 Description

The deviate,

x_{p_{i}}

, associated with the lower tail probability

p_{i}

of the

χ^{2}

-distribution with

ν_{i}

degrees of freedom is defined as the solution to

P (X_{i} \leq x_{p_{i}} : ν_{i}) = p_{i} = \frac{1}{2^{ν_{i} / 2} Γ (ν_{i} / 2)} \int_{0}^{x_{p_{i}}} e^{- X_{i} / 2} {X_{i}}^{v_{i} / 2 - 1} d X_{i}, 0 \leq x_{p_{i}} < \infty; ​ ν_{i} > 0 .

The required

x_{p_{i}}

is found by using the relationship between a

χ^{2}

-distribution and a gamma distribution, 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

For very large values of

ν_{i}

, greater than

10^{5}

, Wilson and Hilferty's Normal approximation to the

χ^{2}

is used; see Kendall and Stuart (1969).

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

Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the

χ^{2}

distribution Appl. Statist. 24 385–388

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin

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 supplied probabilities represent. For

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

, for

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

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

Constraint:

TAIL (j) ='L'

'U'

, for

j = 1, 2, \dots, LTAIL

3: LP – INTEGERInput

On entry: the length of the array P.

Constraint:

LP > 0

4: P(LP) – REAL (KIND=nag_wp) arrayInput

On entry:

p_{i}

, the probability of the required

χ^{2}

-distribution as defined by TAIL with

p_{i} = P (j)

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

Constraints:

if $TAIL (k) ='L'$ , $0.0 \leq P (j) < 1.0$ ;
otherwise $0.0 < P (j) \leq 1.0$ .

Where

k = (i - 1) mod LTAIL + 1

and

j = (i - 1) mod LP + 1

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: X( $*$ ) – REAL (KIND=nag_wp) arrayOutput

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

\max (LTAIL, LP, LDF)

On exit:

x_{p_{i}}

, the deviates for the

χ^{2}

-distribution.

8: IVALID( $*$ ) – INTEGER arrayOutput

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

\max (LTAIL, LP, LDF)

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

x_{p_{i}}

$IVALID (i) = 2$

On entry,

invalid value for

p_{i}

$IVALID (i) = 3$

On entry,

ν_{i} \leq 0.0

$IVALID (i) = 4$

p_{i}

is too close to

0.0

1.0

for the result to be calculated.

$IVALID (i) = 5$

The solution has failed to converge. The result should be a reasonable approximation.

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, 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, P or DF 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: $LP > 0$ .

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

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

7 Accuracy

The results should be accurate to five significant digits for most parameter values. Some accuracy is lost for

p_{i}

close to

0.0

1.0

8 Further Comments

For higher accuracy the relationship described in Section 3 may be used and a direct call to G01TFF made.

9 Example

This example reads lower tail probabilities for several

χ^{2}

-distributions, and calculates and prints the corresponding deviates.

NAG Library Routine DocumentG01TCF

+− 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

G01TCF