stats (version 3.3)

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

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 $P[X \le x]$, otherwise, $P[X > x]$.

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.

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.

Details

The F distribution with df1 = $n_1$ and df2 = $n_2$ degrees of freedom has density $$f(x) = \frac{\Gamma(n_1/2 + n_2/2)}{\Gamma(n_1/2)\Gamma(n_2/2)} \left(\frac{n_1}{n_2}\right)^{n_1/2} x^{n_1/2 -1} \left(1 + \frac{n_1 x}{n_2}\right)^{-(n_1 + n_2) / 2}$$ for $x > 0$.

It is the distribution of the ratio of the mean squares of $n_1$ and $n_2$ 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.

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.

See Also

Distributions for other standard distributions, including dchisq for chi-squared and dt for Student's t distributions.

Examples

## 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