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g01 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_deviates_f_dist (g01fdc)

## 1  Purpose

nag_deviates_f_dist (g01fdc) returns the deviate associated with the given lower tail probability of the $F$ or variance-ratio distribution with real degrees of freedom.

## 2  Specification

 #include #include
 double nag_deviates_f_dist (double p, double df1, double df2, NagError *fail)

## 3  Description

The deviate, ${f}_{p}$, associated with the lower tail probability, $p$, of the $F$-distribution with degrees of freedom ${\nu }_{1}$ and ${\nu }_{2}$ is defined as the solution to
 $P F ≤ fp : ν1 ,ν2 = p = ν 1 12 ν1 ν 2 12 ν2 Γ ν1 + ν2 2 Γ ν1 2 Γ ν2 2 ∫ 0 fp F 12 ν1-2 ν2 + ν1 F -12 ν1 + ν2 dF ,$
where ${\nu }_{1},{\nu }_{2}>0$; $0\le {f}_{p}<\infty$.
The value of ${f}_{p}$ is computed by means of a transformation to a beta distribution, ${P}_{\beta }\left(B\le \beta :a,b\right)$:
 $PF≤f:ν1,ν2=Pβ B≤ν1f ν1f+ν2 :ν1/2,ν2/2$
and using a call to nag_deviates_beta (g01fec).
For very large values of both ${\nu }_{1}$ and ${\nu }_{2}$, greater than ${10}^{5}$, a normal approximation is used. If only one of ${\nu }_{1}$ or ${\nu }_{2}$ is greater than ${10}^{5}$ then a ${\chi }^{2}$ approximation is used; see Abramowitz and Stegun (1972).

## 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:     pdoubleInput
On entry: $p$, the lower tail probability from the required $F$-distribution.
Constraint: $0.0\le {\mathbf{p}}<1.0$.
2:     df1doubleInput
On entry: the degrees of freedom of the numerator variance, ${\nu }_{1}$.
Constraint: ${\mathbf{df1}}>0.0$.
3:     df2doubleInput
On entry: the degrees of freedom of the denominator variance, ${\nu }_{2}$.
Constraint: ${\mathbf{df2}}>0.0$.
4:     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 ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_SOL_NOT_CONV nag_deviates_f_dist (g01fdc) returns $0.0$.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
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 $1.0$. The value of ${f}_{p}$ cannot be computed. This will only occur when the large sample approximations are used.
NE_REAL_ARG_GE
On entry, ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{p}}<1.0$.
NE_REAL_ARG_LE
On entry, ${\mathbf{df1}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{df2}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{df1}}>0.0$ and ${\mathbf{df2}}>0.0$.
NE_REAL_ARG_LT
On entry, ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{p}}\ge 0.0$.
NE_SOL_NOT_CONV
The solution has failed to converge. However, the result should be a reasonable approximation. Alternatively, nag_deviates_beta (g01fec) can be used with a suitable setting of the argument tol.

## 7  Accuracy

The result should be accurate to five significant digits.

## 8  Parallelism and Performance

Not applicable.

For higher accuracy nag_deviates_beta (g01fec) can be used along with the transformations given in Section 3.

## 10  Example

This example reads the lower tail probabilities for several $F$-distributions, and calculates and prints the corresponding deviates until the end of data is reached.

### 10.1  Program Text

Program Text (g01fdce.c)

### 10.2  Program Data

Program Data (g01fdce.d)

### 10.3  Program Results

Program Results (g01fdce.r)