void	nag_deviates_chi_sq_vector (Integer ltail, const Nag_TailProbability tail[], Integer lp, const double p[], Integer ldf, const double df[], double x[], Integer ivalid[], NagError *fail)

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 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

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

Arguments

1: $ltail$ – IntegerInput

On entry: the length of the array tail.

Constraint:

ltail > 0

2: $tail [ltail]$ – const Nag_TailProbabilityInput

On entry: indicates which tail the supplied probabilities represent. For

j = (i - 1) mod ltail

, for

i = 1, 2, \dots, \max (ltail, lp, ldf)

$tail [j] = Nag_LowerTail$: The lower tail probability, i.e., $p_{i} = P (X_{i} \leq x_{p_{i}} : ν_{i})$ .
$tail [j] = Nag_UpperTail$: The upper tail probability, i.e., $p_{i} = P (X_{i} \geq x_{p_{i}} : ν_{i})$ .

Constraint:

tail [j - 1] = Nag_LowerTail

Nag_UpperTail

, for

j = 1, 2, \dots, ltail

3: $lp$ – IntegerInput

On entry: the length of the array p.

Constraint:

lp > 0

4: $p [lp]$ – const doubleInput

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

Constraints:

if $tail [k] = Nag_LowerTail$ , $0.0 \leq p [j] < 1.0$ ;
otherwise $0.0 < p [j] \leq 1.0$ .

Where

k = (i - 1) mod ltail

and

j = (i - 1) mod lp

5: $ldf$ – IntegerInput

On entry: the length of the array df.

Constraint:

ldf > 0

6: $df [ldf]$ – const doubleInput

On entry:

ν_{i}

, the degrees of freedom of the

χ^{2}

-distribution with

ν_{i} = df [j]

j = (i - 1) mod ldf

Constraint:

df [j - 1] > 0.0

, for

j = 1, 2, \dots, ldf

7: $x [\dim]$ – doubleOutput

Note: the dimension, dim, 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: $ivalid [\dim]$ – IntegerOutput

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

\max (ltail, lp, ldf)

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,

invalid value for

p_{i}

$ivalid [i - 1] = 3$

On entry,

ν_{i} \leq 0.0

$ivalid [i - 1] = 4$

p_{i}

is too close to

0.0

1.0

for the result to be calculated.

$ivalid [i - 1] = 5$

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

9: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_ARRAY_SIZE: On entry, $array size = 〈value〉$ .
Constraint: $ldf > 0$ .

On entry, $array size = 〈value〉$ .
Constraint: $lp > 0$ .

On entry, $array size = 〈value〉$ .
Constraint: $ltail > 0$ .
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NW_IVALID: On entry, at least one value of tail, p or df was invalid, or the solution failed to converge.
Check ivalid for more information.

7

Accuracy

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

p_{i}

close to

0.0

1.0

8

Parallelism and Performance

nag_deviates_chi_sq_vector (g01tcc) is not threaded in any implementation.

9

Further Comments

For higher accuracy the relationship described in Section 3 may be used and a direct call to nag_deviates_gamma_vector (g01tfc) made.

10

Example

This example reads lower tail probabilities for several

χ^{2}

-distributions, and calculates and prints the corresponding deviates.

NAG Library Function Document

nag_deviates_chi_sq_vector (g01tcc)

▸▿ 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

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