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g01 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_prob_non_central_f_dist (g01gdc)

## 1  Purpose

nag_prob_non_central_f_dist (g01gdc) returns the probability associated with the lower tail of the noncentral $F$ or variance-ratio distribution.

## 2  Specification

 #include #include
 double nag_prob_non_central_f_dist (double f, double df1, double df2, double lambda, double tol, Integer max_iter, NagError *fail)

## 3  Description

The lower tail probability of the noncentral $F$-distribution with ${\nu }_{1}$ and ${\nu }_{2}$ degrees of freedom and noncentrality parameter $\lambda$, $P\left(F\le f:{\nu }_{1},{\nu }_{2}\text{;}\lambda \right)$, is defined by
 $PF≤f:ν1,ν2;λ=∫0xpF:ν1,ν2;λdF,$
where
 $PF : ν1,ν2;λ =∑j= 0∞e-λ/2 λ/2jj! ×ν1+ 2jν1+ 2j/2 ν2ν2/2 Bν1+ 2j/2,ν2/2$
 $×uν1+2j-2/2ν2+ν1+2ju -ν1+2j+ν2/2$
and $B\left(·,·\right)$ is the beta function.
The probability is computed by means of a transformation to a noncentral beta distribution:
 $PF≤f:ν1,ν2;λ=PβX≤x:a,b;λ,$
where $x=\frac{{\nu }_{1}f}{{\nu }_{1}f+{\nu }_{2}}$ and ${P}_{\beta }\left(X\le x:a,b\text{;}\lambda \right)$ is the lower tail probability integral of the noncentral beta distribution with parameters $a$, $b$, and $\lambda$.
If ${\nu }_{2}$ is very large, greater than ${10}^{6}$, then a ${\chi }^{2}$ approximation is used.
Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

## 5  Arguments

1:     fdoubleInput
On entry: $f$, the deviate from the noncentral $F$-distribution.
Constraint: ${\mathbf{f}}>0.0$.
2:     df1doubleInput
On entry: the degrees of freedom of the numerator variance, ${\nu }_{1}$.
Constraint: $0.0<{\mathbf{df1}}\le {10}^{6}$.
3:     df2doubleInput
On entry: the degrees of freedom of the denominator variance, ${\nu }_{2}$.
Constraint: ${\mathbf{df2}}>0.0$.
On entry: $\lambda$, the noncentrality parameter.
Constraint: $0.0\le {\mathbf{lambda}}\le -2.0\mathrm{log}\left(U\right)$ where $U$ is the safe range parameter as defined by nag_real_safe_small_number (X02AMC).
5:     toldoubleInput
On entry: the relative accuracy required by you in the results. If nag_prob_non_central_f_dist (g01gdc) is entered with tol greater than or equal to $1.0$ or less than  (see nag_machine_precision (X02AJC)), then the value of  is used instead.
6:     max_iterIntegerInput
On entry: the maximum number of iterations to be used.
Suggested value: $500$. See nag_prob_non_central_chi_sq (g01gcc) and nag_prob_non_central_beta_dist (g01gec) for further details.
Constraint: ${\mathbf{max_iter}}\ge 1$.
7:     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_CONV
The solution has failed to converge in $⟨\mathit{\text{value}}⟩$ iterations. Consider increasing max_iter or tol.
NE_INT_ARG_LT
On entry, ${\mathbf{max_iter}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{max_iter}}\ge 1$.
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_PROB_F
The required probability cannot be computed accurately. This may happen if the result would be very close to zero or one. Alternatively the values of df1 and f may be too large. In the latter case you could try using a normal approximation, see Abramowitz and Stegun (1972).
NE_PROB_F_INIT
The required accuracy was not achieved when calculating the initial value of the central $F$ or ${\chi }^{2}$ probability. You should try a larger value of tol. If the ${\chi }^{2}$ approximation is being used then nag_prob_non_central_f_dist (g01gdc) returns zero otherwise the value returned should be an approximation to the correct value.
NE_REAL_ARG_CONS
On entry, ${\mathbf{df1}}=⟨\mathit{\text{value}}⟩$.
Constraint: $0.0<{\mathbf{df1}}\le {10}^{6}$.
On entry, ${\mathbf{df1}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{df1}}>0.0$.
On entry, ${\mathbf{lambda}}=⟨\mathit{\text{value}}⟩$.
Constraint: $0.0\le {\mathbf{lambda}}\le -2.0×\mathrm{log}\left(U\right)$, where $U$ is the safe range parameter as defined by nag_real_safe_small_number (X02AMC).
NE_REAL_ARG_LE
On entry, ${\mathbf{df2}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{df2}}>0.0$.
On entry, ${\mathbf{f}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{f}}>0.0$.

## 7  Accuracy

The relative accuracy should be as specified by tol. For further details see nag_prob_non_central_chi_sq (g01gcc) and nag_prob_non_central_beta_dist (g01gec).

## 8  Parallelism and Performance

Not applicable.

When both ${\nu }_{1}$ and ${\nu }_{2}$ are large a Normal approximation may be used and when only ${\nu }_{1}$ is large a ${\chi }^{2}$ approximation may be used. In both cases $\lambda$ is required to be of the same order as ${\nu }_{1}$. See Abramowitz and Stegun (1972) for further details.

## 10  Example

This example reads values from, and degrees of freedom for, $F$-distributions, computes the lower tail probabilities and prints all these values until the end of data is reached.

### 10.1  Program Text

Program Text (g01gdce.c)

### 10.2  Program Data

Program Data (g01gdce.d)

### 10.3  Program Results

Program Results (g01gdce.r)