naginterfaces.library.rand.dist_​f

naginterfaces.library.rand.dist_f(n, df1, df2, statecomm)

dist_f generates a vector of pseudorandom numbers taken from an $F$ (or Fisher’s variance ratio) distribution with $\mu$ and $\nu$ degrees of freedom.

For full information please refer to the NAG Library document for g05sh

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/g05/g05shf.html

Parameters

nint

$n$, the number of pseudorandom numbers to be generated.

df1int

$\mu$, the number of degrees of freedom of the distribution.

df2int

$\nu$, the number of degrees of freedom of the distribution.

statecommdict, RNG communication object, modified in place

RNG communication structure.

This argument must have been initialized by a prior call to init_repeat() or init_nonrepeat().

Returns

xfloat, ndarray, shape $\left(n\right)$

The $n$ pseudorandom numbers from the specified $F$-distribution.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $2$)

On entry, $df1=⟨value⟩$.

Constraint: $df1\ge 1$.

(errno $3$)

On entry, $df2=⟨value⟩$.

Constraint: $df2\ge 1$.

(errno $4$)

On entry, $statecomm$[‘state’] vector has been corrupted or not initialized.

Notes

The distribution has PDF (probability density function)

$$\begin{array}{c}\begin{array}{cc}f\left(x\right)=\frac{\left(\frac{\mu +\nu -2}{2}\right)!x^{\frac{1}{2}\mu -1}}{(\frac{1}{2}\mu -1)!(\frac{1}{2}\nu -1)!{(1+\frac{\mu }{\nu }x)}^{\frac{1}{2}(\mu +\nu )}}\times {\left(\frac{\mu }{\nu }\right)}^{\frac{1}{2}\mu }& \text{if}x>0\text{,}\\ & \\ f\left(x\right)=0& \text{otherwise.}\end{array}\end{array}$$

dist_f calculates the values

$$\frac{\nu y_i}{\mu z_i}\text{,}i=1,2,\dots ,n\text{,}$$

where $y_i$ and $z_i$ are generated by dist_gamma() from gamma distributions with parameters $(\frac{1}{2}\mu ,2)$ and $(\frac{1}{2}\nu ,2)$ respectively (i.e., from ${\chi }^2$-distributions with $\mu$ and $\nu$ degrees of freedom).

One of the initialization functions init_repeat() (for a repeatable sequence if computed sequentially) or init_nonrepeat() (for a non-repeatable sequence) must be called prior to the first call to dist_f.

References

Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison–Wesley