# Fdist: The F Distribution

## Description

Density, distribution function, quantile function and random generation for the F distribution with `df1` and `df2` degrees of freedom (and optional non-centrality parameter `ncp`).

## Usage

 ```1 2 3 4``` ```df(x, df1, df2, ncp, log = FALSE) pf(q, df1, df2, ncp, lower.tail = TRUE, log.p = FALSE) qf(p, df1, df2, ncp, lower.tail = TRUE, log.p = FALSE) rf(n, df1, df2, ncp) ```

## Arguments

 `x, q` vector of quantiles. `p` vector of probabilities. `n` number of observations. If `length(n) > 1`, the length is taken to be the number required. `df1, df2` degrees of freedom. `Inf` is allowed. `ncp` non-centrality parameter. If omitted the central F is assumed. `log, log.p` logical; if TRUE, probabilities p are given as log(p). `lower.tail` logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x].

## Details

The F distribution with `df1 =` n1 and `df2 =` n2 degrees of freedom has density

f(x) = Γ((n1 + n2)/2) / (Γ(n1/2) Γ(n2/2)) (n1/n2)^(n1/2) x^(n1/2 - 1) (1 + (n1/n2) x)^-(n1 + n2)/2

for x > 0.

It is the distribution of the ratio of the mean squares of n1 and n2 independent standard normals, and hence of the ratio of two independent chi-squared variates each divided by its degrees of freedom. Since the ratio of a normal and the root mean-square of m independent normals has a Student's t_m distribution, the square of a t_m variate has a F distribution on 1 and m degrees of freedom.

The non-central F distribution is again the ratio of mean squares of independent normals of unit variance, but those in the numerator are allowed to have non-zero means and `ncp` is the sum of squares of the means. See Chisquare for further details on non-central distributions.

## Value

`df` gives the density, `pf` gives the distribution function `qf` gives the quantile function, and `rf` generates random deviates.

Invalid arguments will result in return value `NaN`, with a warning.

The length of the result is determined by `n` for `rf`, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than `n` are recycled to the length of the result. Only the first elements of the logical arguments are used.

## Note

Supplying `ncp = 0` uses the algorithm for the non-central distribution, which is not the same algorithm used if `ncp` is omitted. This is to give consistent behaviour in extreme cases with values of `ncp` very near zero.

The code for non-zero `ncp` is principally intended to be used for moderate values of `ncp`: it will not be highly accurate, especially in the tails, for large values.

## Source

For the central case of `df`, computed via a binomial probability, code contributed by Catherine Loader (see `dbinom`); for the non-central case computed via `dbeta`, code contributed by Peter Ruckdeschel.

For `pf`, via `pbeta` (or for large `df2`, via `pchisq`).

For `qf`, via `qchisq` for large `df2`, else via `qbeta`.

## References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 2, chapters 27 and 30. Wiley, New York.

Distributions for other standard distributions, including `dchisq` for chi-squared and `dt` for Student's t distributions.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```## Equivalence of pt(.,nu) with pf(.^2, 1,nu): x <- seq(0.001, 5, len = 100) nu <- 4 stopifnot(all.equal(2*pt(x,nu) - 1, pf(x^2, 1,nu)), ## upper tails: all.equal(2*pt(x, nu, lower=FALSE), pf(x^2, 1,nu, lower=FALSE))) ## the density of the square of a t_m is 2*dt(x, m)/(2*x) # check this is the same as the density of F_{1,m} all.equal(df(x^2, 1, 5), dt(x, 5)/x) ## Identity: qf(2*p - 1, 1, df) == qt(p, df)^2 for p >= 1/2 p <- seq(1/2, .99, length = 50); df <- 10 rel.err <- function(x, y) ifelse(x == y, 0, abs(x-y)/mean(abs(c(x,y)))) quantile(rel.err(qf(2*p - 1, df1 = 1, df2 = df), qt(p, df)^2), .90) # ~= 7e-9 ```