g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_deviates_chi_sq (g01fcc)

## 1  Purpose

nag_deviates_chi_sq (g01fcc) returns the deviate associated with the given lower tail probability of the ${\chi }^{2}$-distribution with real degrees of freedom.

## 2  Specification

 #include #include
 double nag_deviates_chi_sq (double p, double df, NagError *fail)

## 3  Description

The deviate, ${x}_{p}$, associated with the lower tail probability $p$ of the ${\chi }^{2}$-distribution with $\nu$ degrees of freedom is defined as the solution to
 $PX≤xp:ν=p=12ν/2Γν/2 ∫0xpe-X/2Xv/2-1dX, 0≤xp<∞;ν>0.$
The required ${x}_{p}$ is found by using the relationship between a ${\chi }^{2}$-distribution and a gamma distribution, i.e., a ${\chi }^{2}$-distribution with $\nu$ degrees of freedom is equal to a gamma distribution with scale parameter $2$ and shape parameter $\nu /2$.
For very large values of $\nu$, greater than ${10}^{5}$, Wilson and Hilferty's normal approximation to the ${\chi }^{2}$ is used; see Kendall and Stuart (1969).

## 4  References

Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the ${\chi }^{2}$ 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:     pdoubleInput
On entry: $p$, the lower tail probability from the required ${\chi }^{2}$-distribution.
Constraint: $0.0\le {\mathbf{p}}<1.0$.
2:     dfdoubleInput
On entry: $\nu$, the degrees of freedom of the ${\chi }^{2}$-distribution.
Constraint: ${\mathbf{df}}>0.0$.
3:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

On any of the error conditions listed below except NE_ALG_NOT_CONV nag_deviates_chi_sq (g01fcc) returns $0.0$.
NE_ALG_NOT_CONV
The algorithm has failed to converge in $〈\mathit{\text{value}}〉$ iterations. The result should be a reasonable approximation.
NE_GAM_NOT_CONV
The series used to calculate the gamma function has failed to converge. This is an unlikely error exit.
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.
NE_PROBAB_CLOSE_TO_TAIL
The probability is too close to $0.0$ or $1.0$.
NE_REAL_ARG_GE
On entry, ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{p}}<1.0$.
NE_REAL_ARG_LE
On entry, ${\mathbf{df}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{df}}>0.0$.
NE_REAL_ARG_LT
On entry, ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{p}}\ge 0.0$.

## 7  Accuracy

The results should be accurate to five significant digits for most argument values. Some accuracy is lost for $p$ close to $0.0$.

For higher accuracy the relationship described in Section 3 may be used and a direct call to nag_deviates_gamma_dist (g01ffc) made.

## 9  Example

This example reads lower tail probabilities for several ${\chi }^{2}$-distributions, and calculates and prints the corresponding deviates until the end of data is reached.

### 9.1  Program Text

Program Text (g01fcce.c)

### 9.2  Program Data

Program Data (g01fcce.d)

### 9.3  Program Results

Program Results (g01fcce.r)