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

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 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

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

Arguments

1: $ltail$ – IntegerInput

On entry: the length of the array tail.

Constraint:

ltail > 0

2: $tail [ltail]$ – const Nag_TailProbabilityInput

On entry: indicates whether the lower or upper tail probabilities are required. For

j = (i - 1) mod ltail

, for

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

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

Constraint:

tail [j - 1] = Nag_LowerTail

Nag_UpperTail

, for

j = 1, 2, \dots, ltail

3: $lx$ – IntegerInput

On entry: the length of the array x.

Constraint:

lx > 0

4: $x [lx]$ – const doubleInput

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

Constraint:

x [j - 1] \geq 0.0

, for

j = 1, 2, \dots, lx

5: $ldf$ – IntegerInput

On entry: the length of the array df.

Constraint:

ldf > 0

6: $df [ldf]$ – 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: $p [\dim]$ – doubleOutput

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

\max (ltail, ldf, lx)

On exit:

p_{i}

, the probabilities for the

χ^{2}

distribution.

8: $ivalid [\dim]$ – IntegerOutput

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

\max (ltail, ldf, lx)

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,

x_{i} < 0.0

$ivalid [i - 1] = 3$

On entry,

ν_{i} \leq 0.0

$ivalid [i - 1] = 4$

The solution has failed to converge while calculating the gamma variate. The result returned should represent an approximation to the solution.

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: $ltail > 0$ .

On entry, $array size = 〈value〉$ .
Constraint: $lx > 0$ .
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 x, df or tail was invalid, or the solution failed to converge.
Check ivalid for more information.

7

Accuracy

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

8

Parallelism and Performance

nag_prob_chi_sq_vector (g01scc) is not threaded in any implementation.

9

Further Comments

For higher accuracy the transformation described in Section 3 may be used with a direct call to nag_incomplete_gamma (s14bac).

10

Example

Values from various

χ^{2}

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

NAG Library Function Document

nag_prob_chi_sq_vector (g01scc)

▸▿ 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