# NAG CL Interfaceg01fdc (inv_​cdf_​f)

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## 1Purpose

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

## 2Specification

 #include
 double g01fdc (double p, double df1, double df2, NagError *fail)
The function may be called by the names: g01fdc, nag_stat_inv_cdf_f or nag_deviates_f_dist.

## 3Description

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
 $P ( F ≤ fp : ν1 ,ν2) = p = ν 1 12 ν1 ν 2 12 ν2 Γ ( ν1 + ν2 2 ) Γ ( ν1 2 ) Γ ( ν2 2 ) ∫ 0 fp F 12 (ν1-2) (ν2+ν1F) -12 (ν1+ν2) dF ,$
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, ${P}_{\beta }\left(B\le \beta :a,b\right)$:
 $P(F≤f:ν1,ν2)=Pβ (B≤ν1f ν1f+ν2 :ν1/2,ν2/2)$
and using a call to 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).

## 4References

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

## 5Arguments

1: $\mathbf{p}$double Input
On entry: $p$, the lower tail probability from the required $F$-distribution.
Constraint: $0.0\le {\mathbf{p}}<1.0$.
2: $\mathbf{df1}$double Input
On entry: the degrees of freedom of the numerator variance, ${\nu }_{1}$.
Constraint: ${\mathbf{df1}}>0.0$.
3: $\mathbf{df2}$double Input
On entry: the degrees of freedom of the denominator variance, ${\nu }_{2}$.
Constraint: ${\mathbf{df2}}>0.0$.
4: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

If on exit ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_PROBAB_CLOSE_TO_TAIL, NE_REAL_ARG_GE, NE_REAL_ARG_LE or NE_REAL_ARG_LT, then g01fdc returns $0.0$.
On any of the error conditions listed below except ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_SOL_NOT_CONV g01fdc returns $0.0$.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
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, g01fec can be used with a suitable setting of the argument tol.

## 7Accuracy

The result should be accurate to five significant digits.

## 8Parallelism and Performance

g01fdc is not threaded in any implementation.

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

## 10Example

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.

### 10.1Program Text

Program Text (g01fdce.c)

### 10.2Program Data

Program Data (g01fdce.d)

### 10.3Program Results

Program Results (g01fdce.r)