nag_prob_chi_sq_vector (g01scc) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_prob_chi_sq_vector (g01scc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_prob_chi_sq_vector (g01scc) returns a number of lower or upper tail probabilities for the χ2-distribution with real degrees of freedom.

2  Specification

#include <nag.h>
#include <nagg01.h>
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 χ2-distribution with νi degrees of freedom, P = Xi xi :νi  is defined by:
P = Xixi:νi = 1 2 νi/2 Γ νi/2 0.0 xi Xi νi/2-1 e -Xi/2 dXi ,   xi 0 , νi > 0 .
To calculate P = Xi xi :νi  a transformation of a gamma distribution is employed, 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.
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:     ltailIntegerInput
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,,maxltail,lx,ldf:
tail[j]=Nag_LowerTail
The lower tail probability is returned, i.e., pi = P Xi xi :νi .
tail[j]=Nag_UpperTail
The upper tail probability is returned, i.e., pi = P Xi xi :νi .
Constraint: tail[j-1]=Nag_LowerTail or Nag_UpperTail, for j=1,2,,ltail.
3:     lxIntegerInput
On entry: the length of the array x.
Constraint: lx>0.
4:     x[lx]const doubleInput
On entry: xi, the values of the χ2 variates with νi degrees of freedom with xi=x[j], j=i-1 mod lx.
Constraint: x[j-1]0.0, for j=1,2,,lx.
5:     ldfIntegerInput
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:     p[dim]doubleOutput
Note: the dimension, dim, of the array p must be at least maxltail,ldf,lx.
On exit: pi, the probabilities for the χ2 distribution.
8:     ivalid[dim]IntegerOutput
Note: the dimension, dim, of the array ivalid must be at least maxltail,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 pi.
ivalid[i-1]=2
On entry,xi<0.0.
ivalid[i-1]=3
On entry,νi0.0.
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:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
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.
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  Further Comments

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

9  Example

Values from various χ2-distributions are read, the lower tail probabilities calculated, and all these values printed out.

9.1  Program Text

Program Text (g01scce.c)

9.2  Program Data

Program Data (g01scce.d)

9.3  Program Results

Program Results (g01scce.r)


nag_prob_chi_sq_vector (g01scc) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012