NAG Library Function Document

nag_deviates_chi_sq_vector (g01tcc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_deviates_chi_sq_vector (g01tcc) returns a number of deviates associated with the given probabilities of the χ2-distribution with real degrees of freedom.

2
Specification

#include <nag.h>
#include <nagg01.h>
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, xpi, associated with the lower tail probability pi of the χ2-distribution with νi degrees of freedom is defined as the solution to
P Xi xpi :νi = pi = 1 2 νi/2 Γ νi/2 0 xpi e -Xi/2 Xi vi / 2 - 1 dXi ,   0 xpi < ; ​ νi > 0 .  
The required xpi is found by using the relationship between a χ2-distribution and a gamma distribution, i.e., a χ2-distribution with νi degrees of freedom is equal to a gamma distribution with scale parameter 2 and shape parameter νi/2.
For very large values of νi, greater than 105, Wilson and Hilferty's Normal approximation to the χ2 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 χ2 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,,maxltail,lp,ldf:
tail[j]=Nag_LowerTail
The lower tail probability, i.e., pi = P Xi xpi :νi .
tail[j]=Nag_UpperTail
The upper tail probability, i.e., pi = P Xi xpi :νi .
Constraint: tail[j-1]=Nag_LowerTail or Nag_UpperTail, for j=1,2,,ltail.
3:     lp IntegerInput
On entry: the length of the array p.
Constraint: lp>0.
4:     p[lp] const doubleInput
On entry: pi, the probability of the required χ2-distribution as defined by tail with pi=p[j], j=i-1 mod lp.
Constraints:
  • if tail[k]=Nag_LowerTail, 0.0p[j]<1.0;
  • otherwise 0.0<p[j]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: νi, the degrees of freedom of the χ2-distribution with νi=df[j], j=i-1 mod ldf.
Constraint: df[j-1]>0.0, for j=1,2,,ldf.
7:     x[dim] doubleOutput
Note: the dimension, dim, of the array x must be at least maxltail,lp,ldf.
On exit: xpi, the deviates for the χ2-distribution.
8:     ivalid[dim] IntegerOutput
Note: the dimension, dim, of the array ivalid must be at least maxltail,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 xpi.
ivalid[i-1]=2
On entry,invalid value for pi.
ivalid[i-1]=3
On entry,νi0.0.
ivalid[i-1]=4
pi is too close to 0.0 or 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 pi close to 0.0 or 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 χ2-distributions, and calculates and prints the corresponding deviates.

10.1
Program Text

Program Text (g01tcce.c)

10.2
Program Data

Program Data (g01tcce.d)

10.3
Program Results

Program Results (g01tcce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017